(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 584995, 12840]*) (*NotebookOutlinePosition[ 627064, 14264]*) (* CellTagsIndexPosition[ 626989, 14258]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Multidimensional Real Analysis, \n Volumes I and II", FontSize->48], " \n ", StyleBox["by", FontWeight->"Plain"], " \n J.J. Duistermaat and J.A.C. Kolk\n ", StyleBox["Cambridge University Press, 2004", FontWeight->"Plain"] }], "Title", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" code for illustrations, exercises and additional results", FontWeight->"Bold"], " \n by \n \ ", StyleBox["Johan A.C. Kolk\n\n ", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], "Contributions by \n ", StyleBox["W.L.J. van der Kallen (UU) and F. Simons", FontWeight->"Bold"], " ", StyleBox["(TUE)", FontWeight->"Bold"], "\n are gratefully acknowledged " }], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ "At this moment the project is under construction; the goal is to provide \ code for: \n\[FilledCircle] the illustrations in the books,\n\[FilledCircle] \ the exercises in the books,\n\[FilledCircle] additional computations. \ \nThe input originating from various people at different stages of the \ project of writing the books, there is (as yet) no uniformity in the \ construction of the code.\n\nThis code has been tested under", StyleBox[" ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" 5.1", FontWeight->"Bold"], ". " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[CellGroupData[{ Cell["Initialization", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ "The cells below are initialization cells. They are evaluated by ", StyleBox["Mathematica", FontSlant->"Italic"], " as soon as the notebook is openened." }], "Text", CellTags->"UCU311"], Cell[BoxData[ \( (*\ Off[General::"\"]*) \)], "Input", InitializationCell->True, CellTags->"UCU311"], Cell[BoxData[ \( (*\ lightblue = RGBColor[220/256, 248/256, 248/256]; \[IndentingNewLine]defaultplotoptions = Sequence[AxesStyle \[Rule] RGBColor[0, 0.5, 0], \ Background \[Rule] lightbluecolor, ImageSize -> 400]; \[IndentingNewLine]SetOptions[ListPlot, \ Evaluate[defaultplotoptions]]; \[IndentingNewLine]SetOptions[Plot, \ Evaluate[defaultplotoptions]];\ *) \)], "Input", InitializationCell->True, CellTags->"UCU311"] }, Closed]], Cell[CellGroupData[{ Cell["A warning", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(N[\(3 Log[640320]\)\/\@163 - \[Pi]]\), "\[IndentingNewLine]", \(\(3 Log[640320]\)\/\@163 == \[Pi]\), "\[IndentingNewLine]", \(\(3 Log[640320]\)\/\@163 < \[Pi]\), "\[IndentingNewLine]", \(N[\(3 Log[640320]\)\/\@163 - \[Pi], 20]\), "\[IndentingNewLine]", \(N[\@\(\[ExponentialE]\^\(\[Pi]\ \@163\) - 744\)\%3]\), "\ \[IndentingNewLine]", \(N[\@\(\[ExponentialE]\^\(\[Pi]\ \@163\) - 744\)\%3, 20]\), "\[IndentingNewLine]", \(\@\(\[ExponentialE]\^\(\[Pi]\ \@163\) - 744\)\%3 < 640320\), "\[IndentingNewLine]", \(N[\@\(\[ExponentialE]\^\(\[Pi]\ \@163\) - 744\)\%3, 32]\), "\[IndentingNewLine]", \(N[\((\@2\%3\/\@\(27 + 3 \@ 69\)\%3 + \@\(27 + 3 \@ 69\)\%3\/\(3 \@ 2\%3\ \))\)\^27369, 5030] - 248872083860566242801488633985778816168566582615463984666186327177996889\ 794130287696994474581612904561588514301192710192379171399799305891401488394133\ 149658866585963617988675636547948407631504856110204145022057101449742807283745\ 349044713489229346181918805096874878013575556923353742673696224778320245988954\ 021330188348466647046614988940265514373462104020440243949707424358384443518085\ 722840358097062929679889933382659868624398785471672437476033581010058232770325\ 288671140498237982079089990431287680958041449065611648473793797460006654268528\ 910653289074234578398368702750793672907944247393407836016081537881694941536622\ 354795389645788338719703010732492423255860464932719592080734416416940884995001\ 297965439527338534109556225631472247772230281824440018654558291301368411606922\ 994845050838556050237637949150591387757469454306709895023373498752595869449316\ 606578614611429580517061613458015626874196778924457225867320551348551144898211\ 307412861644702494277043219675492384705090308683393258398345621077509284049592\ 628939841220494662289606087429485707665176208596763751080775376705660134601877\ 102706808623385083704763163416133841647181234902568523014554906330744898465446\ 950034570811433400237285702426141033340407021679373188990156358791218198650348\ 893224058833347279226451621964326814419320962988346704587273618997970936633010\ 894468362292302548038860927089257990505837606565437272267338242109959665203275\ 242365597028650587908842357311629984324872399237068185610622882525308195133576\ 360680509731476775600998989424818022668921668871255466030797867764203391743352\ 417770361234623556742805716886286826371548744918786523022395903717847865060788\ 592985252403020060537542636129564913749757990272869378603676720389269941884703\ 497390079248651305070787518472229304676835523411784976227884753642738424032537\ 593171006892800308328208350825894165757110641854633899165463352000712509400393\ 706057751324434941912458367864031043804474171546930765098476298711362565509511\ 334141065951479757321648730858802079297236160479801183695344841506977417032760\ 417642838289903736636796987580383036224461356559232344645741738783654670759079\ 114885744233509780436530814275823779622254137234752634751112415708324257725365\ 486454665346855822606936502156045138577028024350769420624776240097240877505114\ 352882534409438003236821814500906873898893269944000616164741243202139992999892\ 419706344951703777826105570587869104325827129194154676479076870290420281538875\ 595346740295225278662421053721821736218737522433522510077486398910060608503105\ 598718095043357468400950552625647975671614005288806192143795350726970553183450\ 775224485377787284807514966943051424812084340586630542566495883338169528931187\ 327561290381156253168399633972123271079696962459769208482552225913489994456744\ 531614418019149262472389961197753334548229672385129687618298798276361290308183\ 040642825576178936086667478513404282486525031998328974483888813752649419502192\ 715872099804245798709850987624398382552439313031938201589124310129865499387208\ 403484650585370461953198199414358447110283006585773942850787801658598482880852\ 634288703833095348282334660656605533983820063203125994246841462051660690287889\ 829590503732716866139232086149659238449279391592627551020430351364687827471021\ 192779859301117801065439219569499299420368424993003990461640112615325982631980\ 8971152916585811064172283699654029309129460623214205826005262694547534088\)}],\ "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 1: THEORY", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["\<\ Example 1.3.10 New Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(rmin = 0.005;\)\), "\n", \(\(rmax = 1;\)\), "\n", \(\(\[Alpha]min = 0;\)\), "\n", \(\(\[Alpha]max = 2 \[Pi];\)\), "\[IndentingNewLine]", \(\(nr = 89;\)\), "\n", \(\(n\[Alpha] = 155;\)\), "\n", \(g[x1_, x2_] := \(x1\ x2\)\/\(x1\^2 + x2\^2\)\), "\[IndentingNewLine]", \(x1[r_, \[Alpha]_] := r\ Cos[\[Alpha]]\), "\n", \(x2[r_, \[Alpha]_] := r\ Sin[\[Alpha]]\), "\n", \(x3[r_, \[Alpha]_] := g[x1[r, \[Alpha]], x2[r, \[Alpha]]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[{x1[r, \[Alpha]], x2[r, \[Alpha]], x3[r, \[Alpha]]}, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {r, rmin, rmax}, {\[Alpha], \[Alpha]min, \[Alpha]max}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {nr, n\[Alpha]}, ViewPoint \[Rule] {2, 1, 1}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, PlotRange \[Rule] {{\(-1\), 1}, {\(-1\), 1}, {\(-0.5\), 0.5}}, ImageSize \[Rule] 1100];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(Plot3D[g[x1, x2], {x1, \(-1\), 1}, {x2, \(-1\), 1}, PlotPoints \[Rule] 171, PlotRange -> All, ViewPoint \[Rule] {1, 0, 0.8}, ColorFunction \[Rule] Hue, Boxed \[Rule] False, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 1100];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Examples 1.3.11 and 2.3.3 Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["This function ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["g", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" also is studied in Example 2.3.3.", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]] }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(rmin = 0.001;\)\), "\n", \(\(rmax = 1;\)\), "\n", \(\(\[Alpha]min = 0;\)\), "\n", \(\(\[Alpha]max = 2 \[Pi];\)\), "\[IndentingNewLine]", \(\(nr = 89;\)\), "\n", \(\(n\[Alpha] = 155;\)\), "\n", \(g[x1_, x2_] := \(x1\ x2\^2\)\/\(x1\^2 + x2\^4\)\), "\[IndentingNewLine]", \(x1[r_, \[Alpha]_] := r\ Cos[\[Alpha]]\), "\n", \(x2[r_, \[Alpha]_] := r\ Sin[\[Alpha]]\), "\n", \(x3[r_, \[Alpha]_] := g[x1[r, \[Alpha]], x2[r, \[Alpha]]]\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[{x1[r, \[Alpha]], x2[r, \[Alpha]], x3[r, \[Alpha]]}, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {r, rmin, rmax}, {\[Alpha], \[Alpha]min, \[Alpha]max}, \ \[IndentingNewLine]LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {nr, n\[Alpha]}, ViewPoint \[Rule] {\(-5\), 2, 3}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, PlotRange \[Rule] {{\(-1\), 1}, {\(-1\), 1}, {\(-0.5\), 0.5}}, Background \[Rule] lightblue, ImageSize \[Rule] 1100];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 1.7.3", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ "Observe that", StyleBox[" ", FontWeight->"Bold"], StyleBox["Mathematica", FontSlant->"Italic"], " treats the numbers 1.5 and ", Cell[BoxData[ \(TraditionalForm\`3\/2\)]], " differently, they are of the structure Real and Rational, respectively. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(f[a_]\)[x_] := N[\(1\/2\) \((x + a\/x)\), 250]\), "\[IndentingNewLine]", \(Print[\*"\"\\""]\), "\[IndentingNewLine]", \(FixedPointList[f[2], 1.5]\), "\[IndentingNewLine]", \(approx = FixedPointList[f[2], 3\/2]\), "\[IndentingNewLine]", \(Table[ approx[\([k + 1]\)]\^2 - 2 == \(1\/\(4\ approx[\([k]\)]\^2\)\) \((approx[\([k]\)]\^2 - \ 2)\)\^2, {k, Length[approx] - 1}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \ \(Table[ N[approx[\([k]\)] - \@2], {k, Length[approx]}]\), "\[IndentingNewLine]", \(Print[\*"\"\\""]\), "\[IndentingNewLine]", \ \(FixedPoint[f[163], 13]\), "\[IndentingNewLine]", \(% - N[\@163, 250]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 2: THEORY", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["\<\ Example 2.3.5 New Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(g[x1_, x2_] := \(x1\ x2\^2\)\/\(x1\^2 + x2\^4\)\), "\[IndentingNewLine]", \(Together[\[PartialD]\_\(x\_1\)\ g[x\_1, x\_2]]\), "\[IndentingNewLine]", \(Together[\[PartialD]\_\(x\_2\)\ g[x\_1, x\_2]]\), "\[IndentingNewLine]", \(Limit[\[PartialD]\_x1\ g[x1, x2] /. {x1 \[Rule] 0, x2 \[Rule] t}, t \[Rule] 0, Direction \[Rule] \(-1\)]\), "\[IndentingNewLine]", \(Limit[\[PartialD]\_x2\ g[x1, x2] /. {x1 \[Rule] t, x2 \[Rule] t}, t \[Rule] 0, Direction \[Rule] \(-1\)]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Section 2.4 Chain rule ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Using ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" we verify the chain rule ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["D", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["(", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], FontVariations->{"CompatibilityType"->0}], StyleBox["g ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["\[EmptySmallCircle]", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], FontVariations->{"CompatibilityType"->0}], StyleBox[" f", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[")", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], FontVariations->{"CompatibilityType"->0}], StyleBox[" (", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["x", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[") = ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["Dg", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["(", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["f", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["(", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["x", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[")) \[EmptySmallCircle] ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["Df", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["(", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["x", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[") from Theorem 2.4.1. ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]] }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ "For doing so we note that in ", StyleBox["Mathematica", FontSlant->"Italic"], " a vector is just a list, therefore we introduce vectors as variables of \ the form x_List. In ", StyleBox["Mathematica ", FontSlant->"Italic"], StyleBox["t", FontVariations->{"CompatibilityType"->0}], "he definition of the Jacobi matrix of a function ", StyleBox["f", FontSlant->"Italic"], " at the point ", StyleBox["x", FontSlant->"Italic"], " also requires a coordinate system with respect to which the \ differentiations are performed. This is the symbolic vector ", StyleBox["y", FontSlant->"Italic"], " occurring in the following definition. Furthermore, we introduce some \ short notation for vectors that resembles the usual one." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(jacobimatrix[f_, x_List] := \ With[{y = \(Unique[y] &\) /@ x}, Table[\[PartialD]\_\(y[\([j]\)]\)\ \(f[y]\)[\([i]\)], {i, Length[f[y]]}, {j, Length[x]}] /. \ Thread[y \[Rule] x]]\), "\n", \(co[x_, n_] := Table[x\_j, {j, n}]\)}], "Input", FontColor->RGBColor[0.0195315, 0.17969, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["We perform some experiments in order to check that our \ definitions yield the correct results", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], " ", StyleBox["in the case of mappings ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["f", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" : ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^3\)]], "\[RightArrow]", StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^4\)]], " and ", StyleBox["g", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" : ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^4\)]], "\[RightArrow]", StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], "as defined below. Note that we have ", StyleBox["g ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["\[EmptySmallCircle]", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], FontVariations->{"CompatibilityType"->0}], StyleBox[" f : ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^3\)]], "\[RightArrow]", StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], " in this situation" }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(f[{x1_, x2_, x3_}] := {Sin[x1] + Sin[x2] + Sin[x3], Cos[x1] + Cos[x2] + Cos[x3], Tan[x1] + Tan[x2] + Tan[x3], Sin[x1 + x2^2 + x3^3]}\), "\[IndentingNewLine]", \(g[{y1_, y2_, y3_, y4_}] := {y1\ y2, y3 + y4}\), "\[IndentingNewLine]", \(xv = co[x, 3]\), "\[IndentingNewLine]", \(MatrixForm[f[xv]]\), "\[IndentingNewLine]", \(\((j1 = jacobimatrix[f, xv])\) // MatrixForm\)}], "Input"], Cell[BoxData[{ \(yv = co[y, 4]\), "\[IndentingNewLine]", \(MatrixForm[g[yv]]\), "\[IndentingNewLine]", \(MatrixForm[jacobimatrix[g, yv]]\), "\[IndentingNewLine]", \(\((j2 = jacobimatrix[g, f[xv]])\) // MatrixForm\)}], "Input"], Cell[BoxData[ \(\((j3 = jacobimatrix[Function[x, g[f[x]]], xv])\) // MatrixForm\)], "Input"], Cell["Now we are prepared to check whether the chain rule holds. ", "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[ \(j3 == j2 . j1\)], "Input"], Cell[TextData[{ StyleBox["We repeat the experiments in order to check that our definitions \ yield the correct results", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], " ", StyleBox["in the case of mappings ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["f", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" : ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\)]], "\[RightArrow]", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], " and ", StyleBox["g", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" : ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], "\[RightArrow]", StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\)]], " as defined in Exercise 2.19. Note that we have ", StyleBox["g ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["\[EmptySmallCircle]", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], FontVariations->{"CompatibilityType"->0}], StyleBox[" f : ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\)]], "\[RightArrow]", StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\)]], " in this situation." }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(f[{t_}] := {Cos[t], Sin[t]}\), "\[IndentingNewLine]", \(g[{x1_, x2_}] := {x1^\((x2)\)}\), "\[IndentingNewLine]", \(\(xv = {t};\)\), "\[IndentingNewLine]", \(g[f[xv]] // MatrixForm\), "\[IndentingNewLine]", \(\((j3 = jacobimatrix[Function[x, g[f[x]]], xv])\) // MatrixForm\), "\[IndentingNewLine]", \(MatrixForm[f[xv]]\), "\[IndentingNewLine]", \(\((j1 = jacobimatrix[f, xv])\) // MatrixForm\), "\[IndentingNewLine]", \(\(yv = co[y, 2];\)\), "\[IndentingNewLine]", \(MatrixForm[g[yv]]\), "\[IndentingNewLine]", \(MatrixForm[jacobimatrix[g, yv]]\), "\[IndentingNewLine]", \(\((j2 = jacobimatrix[g, f[xv]])\) // MatrixForm\), "\[IndentingNewLine]", \(Simplify[j3 == j2 . j1]\)}], "Input"], Cell[TextData[{ StyleBox["We repeat the experiments in order to check that our definitions \ yield the correct results", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], " ", StyleBox["in the case of mappings ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["f", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" : ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], "\[RightArrow]", StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], " and ", StyleBox["g", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" : ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], "\[RightArrow]", StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], "as defined in Exercise 2.20. Note that we have ", StyleBox["g ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["\[EmptySmallCircle]", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], FontVariations->{"CompatibilityType"->0}], StyleBox[" f : ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], "\[RightArrow]", StyleBox[" ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalR]\^2\)]], " in this situation." }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(f[{x1_, x2_}] := {x1\^2 + x2\^2, Sin[x1\ x2]}\), "\[IndentingNewLine]", \(g[{x1_, x2_}] := {x1\ x2, \[ExponentialE]\^x2}\), "\[IndentingNewLine]", \(\(xv = co[x, 2];\)\), "\[IndentingNewLine]", \(MatrixForm[g[f[xv]]]\), "\[IndentingNewLine]", \(\((j3 = jacobimatrix[Function[x, g[f[x]]], xv])\) // MatrixForm\), "\[IndentingNewLine]", \(MatrixForm[f[xv]]\), "\[IndentingNewLine]", \(\((j1 = jacobimatrix[f, xv])\) // MatrixForm\), "\[IndentingNewLine]", \(\(yv = co[y, 2];\)\), "\[IndentingNewLine]", \(MatrixForm[g[yv]]\), "\[IndentingNewLine]", \(MatrixForm[jacobimatrix[g, yv]]\), "\[IndentingNewLine]", \(\(j2 = jacobimatrix[g, f[xv]];\)\), "\[IndentingNewLine]", \(\((j2 = jacobimatrix[g, f[xv]])\) // MatrixForm\), "\[IndentingNewLine]", \(j3 == j2 . j1\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Section 2.4 Computation of Jacobi determinants", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[StyleBox["At this stage we have all definitions available for \ the computation of Jacobi determinants. ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]]], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[ \(jacobidet[f_, x_List] := Det[jacobimatrix[f, x]]\)], "Input"], Cell[TextData[{ StyleBox["As a verification, we compute the Jacobi determinant of \ diffeomorphisms occurring in several examples and exercises", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], "." }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Off[General::spell1]\ \), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{y1, y2}, {y1, y2} = mapvar; {y1 + y2, y1\/y2}]\), "\[IndentingNewLine]", \(\(mapvar = co[y, 2];\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(Simplify[jacobidet[map, mapvar]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{y1, y2}, {y1, y2} = mapvar; {y1\ y2, \((1 - y1)\) y2}]\), "\[IndentingNewLine]", \(\(mapvar = co[y, 2];\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(jacobidet[map, mapvar]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{x1, x2}, {x1, x2} = mapvar; {Cos[x1] Cosh[x2], Sin[x1] Sinh[x2]}]\), "\[IndentingNewLine]", \(\(mapvar = co[x, 2];\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(Simplify[jacobidet[map, mapvar]]\), "\[IndentingNewLine]", \(\(\(Simplify[% + \(Sin[mapvar[\([1]\)]]\^2\) Cosh[mapvar[\([2]\)]]\^2 + \(Cos[mapvar[\([1]\)]]\^2\) Sinh[mapvar[\([2]\)]]\^2]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(map[mapvar_List] := Block[{r, \[Alpha], \[Theta]}, {r, \[Alpha], \[Theta]} = mapvar; r {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]]}]\), "\[IndentingNewLine]", \(\(mapvar = {r, \[Alpha], \[Theta]};\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(Simplify[jacobidet[map, mapvar]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{y1, y2}, {y1, y2} = mapvar; {Sqrt[y1\ y2], Sqrt[\((1 - y1)\) \((y2 - 1)\)]}]\), "\[IndentingNewLine]", \(\(mapvar = co[y, 2];\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(Simplify[jacobidet[map, mapvar]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{r, \[Alpha], \[Theta]1, \[Theta]2}, {r, \[Alpha], \[Theta]1, \ \[Theta]2} = mapvar; r {Cos[\[Alpha]] Cos[\[Theta]1] Cos[\[Theta]2], Sin[\[Alpha]] Cos[\[Theta]1] Cos[\[Theta]2], Sin[\[Theta]1] Cos[\[Theta]2], Sin[\[Theta]2]}]\), "\[IndentingNewLine]", \(\(mapvar = {r, \[Alpha], \[Theta]\_1, \[Theta]\_2};\)\), "\ \[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(Simplify[jacobidet[map, mapvar]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{y1, y2, y3, y4}, {y1, y2, y3, y4} = mapvar; {y1\ y2\ y3\ y4, \((1 - y1)\) y2\ y3\ y4, \((1 - y2)\) y3\ y4, \((1 - y3)\) y4}]\), "\[IndentingNewLine]", \(\(mapvar = co[y, 4];\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(jacobidet[map, mapvar]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{y1, y2}, {y1, y2} = mapvar; {Sin[y1]\/Cos[y2], Sin[y2]\/Cos[y1]}]\), "\[IndentingNewLine]", \(\(mapvar = co[y, 2];\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(jacobidet[map, mapvar]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{r, \[Alpha]}, {r, \[Alpha]} = mapvar; r {Cos[\[Alpha]], Sin[\[Alpha]]}]\), "\[IndentingNewLine]", \(\(mapvar = {r, \[Alpha]};\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(Simplify[jacobidet[map, mapvar]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{y1, y2, s}, {y1, y2, s} = mapvar; {\((1 - s)\) y1, \((1 - s)\) y2, s\ t}]\), "\[IndentingNewLine]", \(\(mapvar = {y\_1, y\_2, s};\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(Simplify[jacobidet[map, mapvar]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{r, \[Alpha], y3}, {r, \[Alpha], y3} = mapvar; {r\ Cos[\[Alpha]], r\ Sin[\[Alpha]], y3}]\), "\[IndentingNewLine]", \(\(mapvar = {r, \[Alpha], y\_3};\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(Simplify[jacobidet[map, mapvar]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{r, t}, {r, t} = mapvar; {r\ \ x\_1[t], r\ \ x\_2[t]}]\), "\[IndentingNewLine]", \(\(mapvar = {r, t};\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(\(\(jacobidet[map, mapvar]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(map[mapvar_List] := Block[{y1, y2}, {y1, y2} = mapvar; {t\_0 + \[Delta]\ y1, t\_0\^2 + 2 t\_0\ \[Delta]\ y1 + \[Delta]\^2\ y2}]\), \ "\[IndentingNewLine]", \(\(mapvar = co[y, 2];\)\), "\[IndentingNewLine]", \(map[mapvar]\), "\[IndentingNewLine]", \(jacobidet[map, mapvar]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 2.7.1 New Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(g[x1_, x2_] := \(x1\^2 - x2\^2\)\/\(x1\^2 + x2\^2\)\), \ "\[IndentingNewLine]", \(Limit[g[x1, x2], x1 \[Rule] 0, Assumptions \[Rule] {x2 \[NotEqual] 0}]\), "\[IndentingNewLine]", \(Limit[g[x1, x2], x2 \[Rule] 0, Assumptions \[Rule] {x1 \[NotEqual] 0}]\)}], "Input"], Cell[TextData[StyleBox["Notice that the inequality of the mixed partial \ derivatives at the origin cannot be perceived in the illustration with the \ naked eye. In polar coordinates in the plane the qualitative appearance of \ the surface, in a description as a graph, is immediately clear.", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]]], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(rmin = 0.005;\)\), "\n", \(\(rmax = 3;\)\), "\n", \(\(\[Alpha]min = 0;\)\), "\n", \(\(\[Alpha]max = 2 \[Pi];\)\), "\[IndentingNewLine]", \(\(nr = 69;\)\), "\n", \(\(n\[Alpha] = 155;\)\), "\n", \(f[x1_, x2_] := x1\ x2\ g[x1, x2]\), "\[IndentingNewLine]", \(x1[r_, \[Alpha]_] := r\ Cos[\[Alpha]]\), "\n", \(x2[r_, \[Alpha]_] := r\ Sin[\[Alpha]]\), "\n", \(\(\(x3[r_, \[Alpha]_] := f[x1[r, \[Alpha]], x2[r, \[Alpha]]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Simplify[ x3[r, \[Alpha]] \[Equal] \(r\^2\/4\) Sin[4 \[Alpha]]]\), "\[IndentingNewLine]", \(\(\(Plot3D[f[x1, x2], {x1, \(-1\), 1}, {x2, \(-1\), 1}, PlotPoints \[Rule] 171, PlotRange -> All, ViewPoint \[Rule] {1, 1, 0.4}, ColorFunction \[Rule] Hue, Boxed \[Rule] False, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 1100];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(D1f[x1_, x2_] := x2 \( x1\^2 - x2\^2\)\/\(x1\^2 + x2\^2\) + 4 x2 \(\( x1\^2\) x2\^2\)\/\((x1\^2 + x2\^2)\)\^2\), "\ \[IndentingNewLine]", \(D2f[x1_, x2_] := x1 \( x1\^2 - x2\^2\)\/\(x1\^2 + x2\^2\) - 4 x1 \(\( x1\^2\) x2\^2\)\/\((x1\^2 + x2\^2)\)\^2\), "\ \[IndentingNewLine]", \(Simplify[\[PartialD]\_x1 f[x1, x2] \[Equal] D1f[x1, x2]]\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_x2 f[x1, x2] \[Equal] D2f[x1, x2]]\), "\[IndentingNewLine]", \(D1f[0, x\_2]\), "\[IndentingNewLine]", \(D2f[x\_1, 0]\), "\[IndentingNewLine]", \(\[PartialD]\_x2 D1f[0, x2] /. x2 \[Rule] 0\), "\[IndentingNewLine]", \(\[PartialD]\_x1 D2f[x1, 0] /. x1 \[Rule] 0\), "\[IndentingNewLine]", \(%% \[Equal] %\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_x2 D1f[x1, x2] /. {x1 \[Rule] r\ Cos[\[Phi]], x2 \[Rule] r\ Sin[\[Phi]]}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 2.9.9 Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(f[x1_, x2_] := x1\ x2 - x1\^2 - x2\^2 - 2 x1 - 2 x2 + 4;\)\), "\[IndentingNewLine]", \(\(g1 = ContourPlot[f[x1, x2], {x1, \(-4\), 0}, {x2, \(-4\), 0}, \ Contours \[Rule] 100, PlotPoints \[Rule] 300, \ ColorFunction \[Rule] Hue, Ticks \[Rule] None, \ Frame \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(g2 = Graphics[{Thickness[0.002], Line[{{\(-2\), \(-2\)}, {\(-1.25\), \(-1.25\)}}]}];\)\ \), "\ \[IndentingNewLine]", \(\(g3 = Graphics[{Thickness[0.002], Line[{{\(-2\), \(-2\)}, {\(-2\) - 1/\((1.66 Sqrt[2])\), \(-2\) + 1/\((1.66 Sqrt[2])\)}}]}];\)\), "\[IndentingNewLine]", \(\(Show[g1, g2, \ g3, ImageSize \[Rule] 700, DisplayFunction -> $DisplayFunction];\)\)}], "Input"], Cell[TextData[{ StyleBox["The color indicates the level. Notice that the function ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["f ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["changes slowly near its maximum at \[Dash](2,2).", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]] }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 2.10.8 (Frullani's integral) New Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[\(x\^b - x\^a\)\/Log[x], {x, 0, 1}, Assumptions \[Rule] {a > 0, b > a}]\), "\[IndentingNewLine]", \(Integrate[\(\[ExponentialE]\^\(\(-a\)\ t\) - \[ExponentialE]\^\(\(-b\)\ \ t\)\)\/t, {t, 0, \[Infinity]}, Assumptions \[Rule] {a > 0, b > a}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 2.10.14 New Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[Sin[t]\/t, {t, 0, \[Infinity]}]\), "\[IndentingNewLine]", \(Integrate[\(\[ExponentialE]\^\(\(-x\)\ t\)\) Sin[t]\/t, {t, 0, \[Infinity]}, Assumptions \[Rule] {x > 0}]\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_x\ \((% + ArcTan[x])\)]\), "\[IndentingNewLine]", \(%% + ArcTan[x] /. x \[Rule] 1\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 2.10.16 New Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[\(\[ExponentialE]\^\(\(-p\)\ y\)\) \(Cos[a\ y] - Cos[b\ \ y]\)\/y, {y, 0, \[Infinity]}, Assumptions \[Rule] {p > 0, a > 0, b > 0}]\), "\[IndentingNewLine]", \(Integrate[\(Cos[a\ y] - Cos[b\ y]\)\/y, {y, 0, \[Infinity]}, Assumptions \[Rule] {a > 0, b > 0}]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 3: THEORY", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["\<\ Example 3.1.1 (Polar coordinates) New Illustrations \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[StyleBox["In the first illustration the radial variable is \ rendered in exponential form. The second illustration is also useful in \ Exercises 0.1 and 5.36.", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]]], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["The following is time-consuming.", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(\(Clear["\<`*\>"];\)\), "\[IndentingNewLine]", \(gridImage::usage := "\"\ \), "\[IndentingNewLine]", \(\(?gridImage\)\), "\[IndentingNewLine]", \(gridImage[f_, {x_, xmin_, xmax_, xdi_}, {y_, ymin_, ymax_, ydi_}, opts___Rule] := \[IndentingNewLine]Module[{gr}, gr = Union[\[IndentingNewLine]Table[{Hue[ 0.1 + 0.9 \((x - xmin)\)/\((xmax - xmin)\)], Cases[ParametricPlot[f, {y, ymin, ymax}, PlotPoints \[Rule] 200, Compiled \[Rule] False, DisplayFunction \[Rule] Identity], _Line, Infinity]}, {x, xmin, xmax, xdi}], \[IndentingNewLine]Table[{Hue[ 0.1 + 0.9 \((y - ymin)\)/\((ymax - ymin)\)], Cases[ParametricPlot[f, {x, xmin, xmax}, PlotPoints \[Rule] 200, Compiled \[Rule] False, DisplayFunction \[Rule] Identity], _Line, Infinity]}, {y, ymin, ymax, ydi}]]; \[IndentingNewLine]Show[ Graphics[gr, opts, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All, Axes \[Rule] True, ImageSize \[Rule] 801]]]\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ toCart[z_] := {Re[z], Im[z]}\), "\[IndentingNewLine]", \(complexListGridPlot[f_List, more__] := \(gridImage[toCart[#], more] &\) /@ f\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ par[r_, a_] := r\ Exp[I\ a]\), "\[IndentingNewLine]", \(f[z_] := \[ExponentialE]\^z\), "\n", \(\(complexListGridPlot[{{x, y}, f[x + I\ y]}, {x, \(-2\), 2, 0.01}, \ {y, \(-\[Pi]\), \[Pi], 0.01}, Ticks \[Rule] None, Axes \[Rule] False];\)\)}], "Input", FontFamily->"Courier New", FontColor->RGBColor[0.0195315, 0.17969, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(p[\[Alpha]_] := {Cos[\[Alpha]], Sin[\[Alpha]]}\), "\[IndentingNewLine]", \(q[\[Alpha]_] := {\(-1\), 0} + 0.2 p[\[Alpha]]\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[p[\[Alpha]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] darkblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[0.2 p[\[Alpha]], {\[Alpha], 0, \[Pi]/3.5}, PlotStyle -> RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = ParametricPlot[0.22 p[\[Alpha]], {\[Alpha], 0, \[Pi]/3.5}, PlotStyle -> RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot[q[\[Alpha]], {\[Alpha], 0, \[Pi]/7}, PlotStyle -> RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p5 = Graphics[{darkblue, PointSize[0.02], Point[0 p[0]], Point[p[\[Pi]]], Point[p[\[Pi]/3.5]]}];\)\), "\[IndentingNewLine]", \(\(p6 = Graphics[{darkblue, Line[{p[\[Pi]], p[\[Pi]/3.5], 0 p[0], p[0], p[\[Pi]]}]}];\)\), "\[IndentingNewLine]", \(\(Show[p1, p2, p3, p4, p5, p6, Axes \[Rule] False, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, ImageSize \[Rule] 400, DisplayFunction -> $DisplayFunction];\)\)}], "Input"], Cell[TextData[{ "In ", StyleBox["Mathematica", FontSlant->"Italic"], " the functions Re, Im and Arg are numerical, which makes these unfit for \ use in symbolic computations. Therefore new, mathematical, versions of these \ functions are introduced. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(FullSimplify[ Arg[\[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)], {\(-\[Pi]\) < \ \[Alpha] < \[Pi], Im[\[Alpha]] \[Equal] 0}]\), "\[IndentingNewLine]", \(\(\(Simplify[ Arg[Cos[\[Alpha]] + \[ImaginaryI]\ Sin[\[Alpha]]], {\(-\[Pi]\) < \ \[Alpha] < \[Pi], Im[\[Alpha]] \[Equal] 0}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(re[z_] := \(1\/2\) \((z + Conjugate[z])\)\), "\[IndentingNewLine]", \(im[z_] := \(1\/\(2 \[ImaginaryI]\)\) \((z - Conjugate[z])\)\), "\[IndentingNewLine]", \(arg[z_] := 2 ArcTan[ im[z]/\((\@\(re[z]\^2 + im[z]\^2\) + re[z])\)]\), "\[IndentingNewLine]", \(Simplify[ r\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\), {r > 0, Im[\[Alpha]] \[Equal] 0}]\), "\[IndentingNewLine]", \(Simplify[ Conjugate[r\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)], {r > 0, Im[\[Alpha]] \[Equal] 0}]\), "\[IndentingNewLine]", \(FullSimplify[re[\[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)], Im[\[Alpha]] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[im[\[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)], Im[\[Alpha]] \[Equal] 0]\), "\[IndentingNewLine]", \(Simplify[ arg[x + \[ImaginaryI]\ y], {Im[x] \[Equal] 0, Im[y] \[Equal] 0}]\), "\[IndentingNewLine]", \(\(\(FullSimplify[ arg[\[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)], {\(-\[Pi]\) < \ \[Alpha] < \[Pi], Im[\[Alpha]] \[Equal] 0}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Simplify[Sin[\[Alpha]]/\((1 + Cos[\[Alpha]])\)]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 3.3.A Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Formulae have been added by means of Xfig.", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(\(Clear["\<`*\>"];\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\[IndentingNewLine]", \(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-2\)\ h} + {x, y}, R[a] . {0, 2\ h} + {x, y}, R[a] . {2\ b, \(-2\)\ h} + {x, y}, R[a] . {0, \(-h\)} + {x, y}}]\), "\n", \(\(Plot[ 0.5/x, {x, 0.5, 3.5}, \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[0.5]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[ 2/x, {x, 1.25, 3.75}, \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[0.5]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[Sqrt[x^2 - 1], {x, 1, 2}, \ \ \ \ \ \ \ PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[0.5]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[Sqrt[x^2 - 9], {x, 3, 3.25}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[0.5]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[ Sqrt[x^2 - 1], {x, Sqrt[0.5 + 0.5\ Sqrt[2]], \ \ Sqrt[0.5 + 0.5\ Sqrt[17]]}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[1.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[ Sqrt[x^2 - 9], {x, Sqrt[4.5 + 0.5\ Sqrt[82]], Sqrt[4.5 + 0.5\ Sqrt[97]]}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[1.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[0.5/x, {x, Sqrt[0.5 + 0.5\ Sqrt[2]], Sqrt[4.5 + 0.5\ Sqrt[82]]}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[1.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[ 2/x, \ \ {x, Sqrt[0.5 + 0.5\ Sqrt[17]], Sqrt[4.5 + 0.5\ Sqrt[97]]}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[1.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{Text["\", {0.5, 1.2}], Text["\", {1, 1.7}], Text["\", {2.1, 1.7}], Text["\", {3, 1.4}], Text["\", {2, 0.5}]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[%%%%%%%%%, %%%%%%%%, %%%%%%%, %%%%%%, %%%%%, %%%%, %%%, %%, %, AspectRatio \[Rule] 0.8, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Ticks \[Rule] {{1, 3}, {1}}, PlotRange \[Rule] {\(-0.6\), 2}, DefaultFont \[Rule] 12. , Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[ Graphics[{RGBColor[0, 0.5, 0], {AbsoluteThickness[0.5], Line[{{\(-4\), 2.5}, {\(-1.5\), 2.5}}], Line[{{\(-1\), 0}, {10, 0}}], Line[{{0, \(-1\)}, {0, 5}}]}, {RGBColor[1, 0, 0], AbsoluteThickness[0.5], Line[{{0.5, 1}, {9.5, 1}}], Line[{{0.5, 4}, {9.5, 4}}], Line[{{1, 0.5}, {1, 4.5}}], Line[{{9, 0.5}, {9, 4.5}}]}, {darkblue, arrow[\(-1.5\), 2.5, 0.2, 0.1, 0.5\ \[Pi]]}, {RGBColor[1, 0, 0], AbsoluteThickness[1.75], Line[{{1, 1}, {9, 1}, {9, 4}, {1, 4}, {1, 1}}]}, \[IndentingNewLine]Text["\", {5, 2.5}], Text["\<1\>", {1, \(-0.6\)}], Text["\<9\>", {9, \(-0.6\)}], Text["\<1\>", {\(-0.45\), 1}], Text["\<4\>", {\(-0.45\), 4}]}], DefaultFont \[Rule] 12. , PlotRange \[Rule] {{\(-4\), 10}, {\(-1\), 5}}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[GraphicsArray[{%%, %}], GraphicsSpacing \[Rule] 0, ImageSize \[Rule] 900, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 3.3.B Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Formulae have been added by means of Xfig.", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(\(Clear["\<`*\>"];\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(t = 0.75\ \[Pi];\)\), "\n", \(\(d = 0.15\ \[Pi];\)\), "\n", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-4\)\ h} + {x, y}, {x, y}, R[a] . {2\ b, \(-4\)\ h} + {x, y}, R[a] . {0, \(-3\)\ h} + {x, y}}]\), "\[IndentingNewLine]", \(\(Graphics[{Text["\", {2\ Cos[t], Sin[t]}, {1.5, \(-1.5\)}], Text["\", {\(-2\)\ Sin[t], Cos[t]}, {1, 2}], Text["\<0\>", {0, 0}, {\(-4\), 1}], Circle[{0, 0}, {2, 1}, {0.4\ \[Pi], 1.2\ \[Pi]}], {AbsoluteThickness[2], Line[{{0, 0}, {2\ Cos[t], Sin[t]}}], Line[{{0, 0}, {\(-2\)\ Sin[t], Cos[t]}}]}, {AbsoluteThickness[1], Line[{{0, 0}, {2\ Cos[t + d], Sin[t + d]}}], Line[{{0, 0}, {2\ Cos[t - d], Sin[t - d]}}], Table[Circle[{0, 0}, {2\ r/10, r/10}, {t - d, t + d}], {r, 1, 10}]}, arrow[2\ Cos[t], Sin[t], 0.025, 0.025, ArcTan[t - 0.5\ \[Pi]] - 0.55\ \[Pi]], arrow[\(-2\)\ Sin[t], Cos[t], 0.025, 0.025, ArcTan[t] + \[Pi]]}, DefaultColor \[Rule] darkblue, Background \[Rule] lightblue, AspectRatio \[Rule] Automatic];\)\), "\n", \(\(Show[%, DefaultFont \[Rule] 12. , ImageSize \[Rule] 500];\)\)}], "Input"], Cell[BoxData[{ \(\(Clear["\<`*\>"];\)\), "\[IndentingNewLine]", \(a[t_] := ArcTan[x\_2[t]\/x\_1[t]]\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_t\ a[t]]\), "\[IndentingNewLine]", \(aa[t_] := 2 ArcTan[ x\_2[t]\/\(\@\(x\_1[t]\^2 + x\_2[t]\^2\) + x\_1[t]\)]\), "\ \[IndentingNewLine]", \(FullSimplify[\[PartialD]\_t\ aa[t]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Application 3.6.B", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[ General::spell1]\), "\[IndentingNewLine]", \(jacobimatrix[f_, x_List] := With[{y = \(Unique[y] &\) /@ x}, Table[\[PartialD]\_\(y[\([j]\)]\)\ \(f[y]\)[\([i]\)], {i, Length[f[y]]}, {j, Length[x]}] /. \ Thread[y \[Rule] x]]\), "\[IndentingNewLine]", RowBox[{ " ", \(co[x_, n_] := Table[x\_j, {j, n}]\)}], "\[IndentingNewLine]", \(Unprotect[ Power];\), "\n", RowBox[{\(Format[Power[Subscript[x_, y_], z_]] := DisplayForm[SuperscriptBox[SubscriptBox[x, y], z]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(f[x_, y_List] := \(x\^2\) y[\([1]\)] + \[ExponentialE]\^\(2 x\) + y[\([2]\)]\), "\[IndentingNewLine]", \(yv = co[y, 2]\), "\[IndentingNewLine]", \(f[x, yv]\), "\[IndentingNewLine]", \(x0 = 0;\), "\[IndentingNewLine]", \(y0 = {1, \(-1\)};\), "\ \[IndentingNewLine]", \(f[x0, y0]\), "\[IndentingNewLine]", \(Dxf = \[PartialD]\_x\ f[x, y0] /. x \[Rule] x0\), "\[IndentingNewLine]", \(fx[ y_List] := {f[x0, y]}\), "\[IndentingNewLine]", \(fx[ yv]\), "\[IndentingNewLine]", \(Dyf = jacobimatrix[fx, y0];\), "\[IndentingNewLine]", \(MatrixForm[ Dyf]\), "\[IndentingNewLine]", \(MatrixForm[\(-Dxf^\((\(-1\))\)\) Dyf]\), "\[IndentingNewLine]", \(fy[y1_, y2_] := f[x[y1, y2], {y1, y2}]\), "\[IndentingNewLine]", \(Dy1f = \[PartialD]\_y1\ fy[y1, y2];\), "\[IndentingNewLine]", \(Dy2f = \[PartialD]\_y2\ fy[y1, y2];\), "\[IndentingNewLine]", RowBox[{"Dy1f", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}], "\[Rule]", "\"\<(\!\(D\_1\)x)\>\""}], ",", \(x[y1, y2] -> x\), ",", \(y1 \[Rule] y\_1\)}], "}"}]}], "\[IndentingNewLine]", RowBox[{"Dy2f", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}], "\[Rule]", "\"\<(\!\(D\_2\)x)\>\""}], ",", \(x[y1, y2] -> x\), ",", \(y1 \[Rule] y\_1\)}], "}"}]}], "\[IndentingNewLine]", RowBox[{"Flatten", "[", RowBox[{ RowBox[{ RowBox[{"Solve", "[", RowBox[{\(Dy1f \[Equal] 0\), ",", RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}]}], "]"}], "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}], "\[Rule]", "\"\<\!\(D\_1\)x(1,-1)\>\""}], ",", \(x[y1, y2] \[Rule] 0\), ",", \(y1 \[Rule] 1\), ",", \(y2 \[Rule] \(-1\)\)}], "}"}]}], ",", "2"}], "]"}], "\[IndentingNewLine]", RowBox[{"First", "[", RowBox[{ RowBox[{"Flatten", "[", RowBox[{ RowBox[{ RowBox[{"Solve", "[", RowBox[{\(Dy2f \[Equal] 0\), ",", RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}]}], "]"}], "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}], "\[Rule]", "\"\<\!\(D\_2\)x(1,-1)\>\""}], ",", \(x[y1, y2] \[Rule] 0\), ",", \(y1 \[Rule] 1\), ",", \(y2 \[Rule] \(-1\)\)}], "}"}]}], ",", "2"}], "]"}], "/.", \({Rule \[Rule] Equal}\)}], "]"}], "\[IndentingNewLine]", \(Dyy2f = \[PartialD]\_\(y2, y2\)fy[y1, y2];\), "\[IndentingNewLine]", RowBox[{"Dyy2f", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}], "\[Rule]", "\"\<(\!\(D\_2\^2\)x)\>\""}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}], "\[Rule]", "\"\<(\!\(D\_2\)x)\>\""}], ",", \(x[y1, y2] -> x\), ",", \(y1 \[Rule] y\_1\)}], "}"}]}], "\[IndentingNewLine]", RowBox[{"First", "[", RowBox[{ RowBox[{"Flatten", "[", RowBox[{ RowBox[{ RowBox[{"Solve", "[", RowBox[{\(Dyy2f \[Equal] 0\), ",", RowBox[{ SuperscriptBox["x", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}]}], "]"}], "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}], "\[Rule]", "\"\<\!\(D\_2\^2\)x(1,-1)\>\""}], ",", RowBox[{ RowBox[{ SuperscriptBox["x", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(y1, y2\), "]"}], "\[Rule]", \(\(-1\)/2\)}], ",", \(x[y1, y2] \[Rule] 0\), ",", \(y1 \[Rule] 1\), ",", \(y2 \[Rule] \(-1\)\)}], "}"}]}], ",", "2"}], "]"}], "/.", \({Rule \[Rule] Equal}\)}], "]"}]}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 4: THEORY", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Section 4.1 Introductory remarks", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Print[Helix]\), "\[IndentingNewLine]", \(\(view = {18, 3, 0.5};\)\), "\[IndentingNewLine]", \(\(\(ParametricPlot3D[ Append[{w, Cos[w], Sin[w]}, RGBColor[1, 0, 0]] // Evaluate, {w, \(-6.1\) \[Pi], 7.3 \[Pi]}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, Ticks \[Rule] None, AxesEdge \[Rule] {{\(-1\), \(-1\)}, {\(-1\), \(-1\)}, {\(-1\), \ \(-1\)}}, PlotPoints \[Rule] 200, ViewPoint \[Rule] view, Background \[Rule] lightblue, ImageSize \[Rule] 500];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(<< Graphics`ImplicitPlot`\), "\[IndentingNewLine]", \(\(p1 = ImplicitPlot[x1\^2\/5 + x2\^2\/2 == 1, {x1, \(-3\), 3}, Ticks \[Rule] None, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ImplicitPlot[x1\^2\/3 - x2\^2\/2 == 1, {x1, \(-5\), 5}, Ticks \[Rule] None, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = ImplicitPlot[x1\/3 + x2\^2\/2 \[Equal] 1, {x1, \(-1\), 3}, Ticks \[Rule] None, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(\(Show[GraphicsArray[{p1, p2, p3}], ImageSize \[Rule] 1200, DisplayFunction -> $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print[\*"\"\\""]\), "\[IndentingNewLine]", \(x1[\[Alpha]_, t_] := Cosh[t] Cos[\[Alpha]]\), "\n", \(x2[\[Alpha]_, t_] := Cosh[t] Sin[\[Alpha]]\), "\n", \(x3[\[Alpha]_, t_] := Sinh[t]\), "\[IndentingNewLine]", \(Simplify[ x1[\[Alpha], t]\^2 + x2[\[Alpha], t]\^2 - x3[\[Alpha], t]\^2]\), "\[IndentingNewLine]", \(\(x[\[Alpha]_, t_] := {x1[\[Alpha], t], x2[\[Alpha], t], x3[\[Alpha], t]};\)\), "\[IndentingNewLine]", \(\(h = ParametricPlot3D[ Append[x[\[Alpha], t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {t, \(-1.5\), 1.5}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {121, 121}, Axes \[Rule] False, PlotRange \[Rule] All, BoxRatios \[Rule] {1, 1, 1}, Boxed \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(view = {2, 2, 1};\)\), "\[IndentingNewLine]", \(\(Show[h, PlotRange \[Rule] All, Axes \[Rule] False, BoxRatios \[Rule] {1, 1, 1}, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] view, ImageSize \[Rule] 800, DisplayFunction -> $DisplayFunction];\)\ \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(Print[\*"\"\\""]\), "\n", \(x1[\[Alpha]_, t_] := t\ Cos[\[Alpha]]\), "\n", \(x2[\[Alpha]_, t_] := t\ Sin[\[Alpha]]\), "\n", \(x3[\[Alpha]_, t_] := t\^2\), "\n", \(Simplify[\(-x1[\[Alpha], t]\^2\) - x2[\[Alpha], t]\^2 + x3[\[Alpha], t]]\), "\n", \(\(x[\[Alpha]_, t_] := {x1[\[Alpha], t], x2[\[Alpha], t], x3[\[Alpha], t]};\)\), "\n", \(\(h = ParametricPlot3D[ Append[x[\[Alpha], t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {t, 0, 3}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {121, 121}, Axes \[Rule] False, PlotRange \[Rule] All, BoxRatios \[Rule] {1, 1, 1}, Boxed \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(view = {2, 2, 1};\)\), "\n", \(\(Show[h, PlotRange \[Rule] All, Axes \[Rule] False, BoxRatios \[Rule] {1, 1, 1}, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] view, ImageSize \[Rule] 800, DisplayFunction -> $DisplayFunction];\)\ \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(Print[\*"\"\\""]\), "\n", \(<< Graphics`ContourPlot3D`\), "\n", \(\(ContourPlot3D[ x1\^2 - x2\^2 + x3, {x1, \(-4\), 4}, {x2, \(-4\), 4}, {x3, \(-4\), 4}, PlotPoints \[Rule] 3 {3, 4}, PlotRange \[Rule] {{\(-3\), 3}, {\(-2\), 2}, {\(-2\), 2}}, ViewPoint \[Rule] {2, 1.4, 1.6}, \[IndentingNewLine]ContourStyle \[Rule] RGBColor[0.7, 0.7, 1], LightSources \[Rule] {{{1, 0, 0}, RGBColor[0, 0, 1]}, {{0, 1, 0}, RGBColor[0, 0, 1]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, Boxed \[Rule] False, ImageSize \[Rule] 800];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Definition 4.2.1 Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["This has been produced using Xfig.", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Section 4.3 Immersion Theorem", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(r = 3;\)\), "\n", \(\(s = 1.2;\)\), "\n", \(\(t0 = 0.16\ \[Pi];\)\), "\n", \(\(t1 = 0.35\ \[Pi];\)\), "\n", \(\(t3 = 0.22\ \[Pi];\)\), "\n", \(\(t4 = 0.27\ \[Pi];\)\), "\n", \(\(x = 0.1;\)\), "\n", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(\(\(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-2\)\ h} + {x, y}, R[a] . {0, 2\ h} + {x, y}, R[a] . {2\ b, \(-2\)\ h} + {x, y}, R[a] . {0, \(-h\)} + {x, y}}]\)\(\n\) \)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[{r\ Cos[t], 1 + s\ Sin[t]}, {t, \(-0.2\)\ \[Pi], 0.4\ \[Pi]}, PlotPoints \[Rule] 50, PlotStyle \[Rule] AbsoluteThickness[1], DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = ParametricPlot[{r\ Cos[t], 1 + s\ Sin[t]}, {t, t0, t1}, PlotPoints \[Rule] 50, PlotStyle \[Rule] AbsoluteThickness[2], DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = Graphics[{{AbsolutePointSize[5], Point[{r\ Cos[t4], 0.6}], Point[{r\ Cos[t3], 1 + s\ Sin[t3]}], Point[{r\ Cos[t4], 1 + s\ Sin[t4]}]}, {AbsoluteThickness[1], Dashing[{0.01, 0.01}], Line[{{r\ Cos[t4], 0.6}, {r\ Cos[t4], 1 + s\ Sin[t4]}, {0.5, 1 + s\ Sin[t4]}}]}, {AbsoluteThickness[3], Line[{{r\ Cos[t0], 0.6}, {r\ Cos[t1], 0.6}}]}, {AbsoluteThickness[1], Line[{{0.25, 0.6}, {3.2, 0.6}}], Line[{{0.5, 0.4}, {0.5, 2.4}}]}}];\)\), "\n", \(\(p4 = Graphics[{{AbsolutePointSize[5], Point[{4.9, 0.6}]}, {AbsoluteThickness[3], Line[{{4.25, 0.6}, {5.5, 0.6}}]}, {AbsoluteThickness[1], Line[{{4, 0.4}, {4, 2.4}}], Line[{{3.75, 0.6}, {6, 0.6}}]}}];\)\), "\n", \(\(p5 = Graphics[{{AbsolutePointSize[4], Point[{2.25, \(-0.75\)}]}, {AbsolutePointSize[6], Point[{3, \(-0.75\)}], Point[{3.5, \(-0.75\)}]}, {AbsoluteThickness[3], Line[{{2.5, \(-0.75\)}, {4, \(-0.75\)}}]}, {AbsoluteThickness[ 1], Line[{{1.5, \(-0.75\)}, {1.965, \(-0.75\)}}], Line[{{2.035, \(-0.75\)}, {4.465, \(-0.75\)}}], Line[{{4.535, \(-0.75\)}, {5, \(-0.75\)}}], Circle[{2, \(-0.75\)}, 0.035], Circle[{4.5, \(-0.75\)}, 0.035]}}];\)\), "\n", \(\(p6 = Graphics[{{AbsoluteThickness[1], Line[{{3, 1.5}, {3.75, 1.15}}], Line[{{2.25, 0.25}, {2.75, \(-0.4\)}}], Line[{{2.6, 0.2}, {2.9, \(-0.5\)}}]}, arrow[3.75, 1.15, 0.04, 0.03, 0.5\ \[Pi] + ArcTan[0.35/0.65]], arrow[2.75, \(-0.4\), 0.04, 0.03, 0.5\ \[Pi] + ArcTan[0.65/0.5]], arrow[2.6, 0.2, 0.04, 0.03, ArcTan[0.7/0.3] - 0.5\ \[Pi]]}];\)\), "\n", \(\(p7 = Graphics[{Text["\", {2.25, \(-1\)}], Text["\", {3, \(-1\)}], Text["\", {3.5, \(-1\)}], Text["\", {3.6, \(-0.5\)}], Text["\", {4.3, \(-0.5\)}], Text["\", {5.5, \(-0.75\)}], Text["\", {3.7, 0.4}], Text["\", {4.95, 0.4}], Text["\", {6, 0.4}], Text["\", {4.2, 2.3}], Text["\", {0.3, 0.4}], Text["\", {0.4, 1.9}], Text["\", {0.6, 2.3}], Text["\", {1.5, 0.75}], Text["\", {2, 0.4}], Text["\", {3.1, 1}], Text["\", {2.4, 1.9}], Text["\", {2, 2.1}], Text["\", {3.4, 0.4}], Text["\", {3.5, 1.5}], Text["\", {2.4, \(-0.3\)}], Text["\", {3, 0}]}];\)\), "\[IndentingNewLine]", \(\(Show[p1, p2, p3, p4, p5, p6, p7, PlotRange \[Rule] {{0.2, 6.2}, {\(-1\), 2.4}}, AxesOrigin \[Rule] {1, 0.6}, Axes \[Rule] None, AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, ImageSize \[Rule] 900, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Corollary 4.3.2 Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\n", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-4\)\ h} + {x, y}, {x, y}, R[a] . {2\ b, \(-4\)\ h} + {x, y}, R[a] . {0, \(-3\)\ h} + {x, y}}]\), "\n", \(\(r0 = 2;\)\), "\n", \(\(s0 = 1;\)\), "\n", \(r[t_] := r0 + 2\ \[ExponentialE]\^\(-t\^2\)\), "\n", \(s[t_] := s0 + 3\ \[ExponentialE]\^\(-t\)\), "\n", \(\(t0 = 2.8\ \[Pi];\)\), "\n", \(\(t1 = 0.925\ \[Pi];\)\), "\n", \(\(p1 = ParametricPlot[{\(-r[t]\)\ Cos[t] - Sin[t], \(-s[t]\)\ Sin[t]}, {t, \(-0.4\)\ \[Pi], t0}, PlotStyle \[Rule] RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = ParametricPlot[{\(-r[t]\)\ Cos[t] - Sin[t], \(-s[t]\)\ Sin[t]}, {t, 2.75\ \[Pi], 2.9\ \[Pi]}, PlotStyle \[Rule] {RGBColor[1, 0, 0], Dashing[{0.01, 0.01}]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = Graphics[{{darkblue, Line[{{\(-4.5\), \(-3.5\)}, {2.5, \(-3.5\)}}]}, {darkblue, Line[{{1, \(-1.5\)}, {0.1, \(-2.85\)}}]}, {darkblue, AbsolutePointSize[5], Point[{\(-r[t0]\)\ Cos[t0] - Sin[t0], \(-s[t0]\)\ Sin[t0]}], Point[{\(-r[t1]\)\ Cos[t1] - Sin[t1], \(-s[t1]\)\ Sin[t1]}], Point[{\(-3\), \(-3.5\)}], Point[{2, \(-3.5\)}]}, {darkblue, arrow[0, \(-3\), 0.075, 0.05, ArcTan[\(-1.5\)] - 0.5\ \[Pi]]}, Text["\", {\(-r[t0]\)\ Cos[t0] - Sin[t0], \(-s[t0]\)\ Sin[t0]}], Text["\", {\(-r[t1]\)\ Cos[t1] - Sin[t1], \(-s[t1]\)\ Sin[t1]}], Text["\", {\(-3\), \(-3.5\)}], Text["\", {2, \(-3.5\)}], Text["\", {0, \(-3\)}]}];\)\), "\n", \(\(Show[p1, p2, p3, AspectRatio \[Rule] 1, Axes \[Rule] None, PlotRange \[Rule] {\(-4\), 3}, DefaultFont \[Rule] 15. , Background \[Rule] lightblue, ImageSize \[Rule] 400, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 4.4.2 Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\[CurlyPhi][t_] := {t\^2 - 1, t\^3 - t}\), "\[IndentingNewLine]", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(\(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-4\)\ h} + {x, y}, {x, y}, R[a] . {2\ b, \(-4\)\ h} + {x, y}, R[a] . {0, \(-3\)\ h} + {x, y}}];\)\), "\n", \(\(p1 = ParametricPlot[\[CurlyPhi][t], {t, \(-2\), 1}, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[1.5]}, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 50, Ticks \[Rule] None, PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1\), 1}}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = ParametricPlot[\[CurlyPhi][t], {t, 1, 2}, PlotStyle \[Rule] {RGBColor[1, 0, 0], Dashing[{0.025, 0.025}]}, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 50, Ticks \[Rule] None, PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1\), 1}}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = Graphics[{darkblue, {arrow[0, 0, 0.03, 0.025, 0.25\ \[Pi]], arrow[0.5, \(-Sqrt[3/2]^3\) + Sqrt[3/2], 0.03, 0.025, \(-0.175\)\ \[Pi]]}}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[p1, p2, p3, AxesStyle \[Rule] RGBColor[0, 0.5, 0], ImageSize \[Rule] 600, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 4.4.3 (Torus) Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Missing. p343V", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Section 4.5 Submersion Theorem", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(f[t_, c_] := \(-0.75\)\ \((t - 1)\)\^2 + c\), "\[IndentingNewLine]", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-2\)\ h} + {x, y}, R[a] . {0, 2\ h} + {x, y}, R[a] . {2\ b, \(-2\)\ h} + {x, y}, R[a] . {0, \(-h\)} + {x, y}}]\), "\n", \(\(t0 = 2.89297;\)\), "\n", \(\(t1 = 3.75379;\)\), "\n", \(\(t2 = 1.25;\)\), "\n", \(\(t3 = 3.01556;\)\), "\n", \(\(c0 = \(-3.15\);\)\), "\n", \(\(c = \(-2.6\);\)\), "\n", \(\(y0 = 2.4;\)\), "\n", \(\(y = 3;\)\), "\n", \(\(p1 = ParametricPlot[{f[t, 6], t - 0.5}, {t, \(-0.15\), 2.65201}, PlotPoints \[Rule] 200, PlotStyle \[Rule] {AbsoluteThickness[0.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = ParametricPlot[{f[t, 6], t - 0.5}, {t, 2.65201, 3.5}, PlotPoints \[Rule] 200, PlotStyle \[Rule] {AbsoluteThickness[2]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = ParametricPlot[{f[t, 6], t - 0.5}, {t, 3.5, 3.65}, PlotPoints \[Rule] 200, PlotStyle \[Rule] {AbsoluteThickness[0.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p4 = ParametricPlot[{f[t, 4.5], t - 0.5}, {t, \(-0.15\), 1.8539}, PlotPoints \[Rule] 200, PlotStyle \[Rule] {AbsoluteThickness[0.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p5 = ParametricPlot[{f[t, 4.5], t - 0.5}, {t, 1.8539, 3.06156}, PlotPoints \[Rule] 200, PlotStyle \[Rule] {AbsoluteThickness[2]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p6 = ParametricPlot[{f[t, 4.5], t - 0.5}, {t, 3.06156, 3.3}, PlotPoints \[Rule] 200, PlotStyle \[Rule] {AbsoluteThickness[0.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p7 = ParametricPlot[{f[t, 7], t - 0.5}, {t, t3, t1}, PlotPoints \[Rule] 200, PlotStyle \[Rule] {AbsoluteThickness[0.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p8 = ParametricPlot[{f[t, 4], t - 0.75}, {t, t2, t0}, PlotPoints \[Rule] 200, PlotStyle \[Rule] {AbsoluteThickness[0.75]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p9 = Graphics[{{AbsoluteThickness[0.5], Line[{{\(-1\), 0}, {6.5, 0}}], Line[{{0, \(-0.5\)}, {0, 3.75}}]}, {AbsoluteThickness[0.75], Line[{{f[t0, 4], t0 - 0.75}, {f[t1, 7], t1 - 0.5}}], Line[{{f[t2, 4], t2 - 0.75}, {f[t3, 7], t3 - 0.5}}]}, {AbsoluteThickness[0.75], Dashing[{0.01, 0.01}], Line[{{y, 0}, {y, 2.5}}], Line[{{y0, 0}, {y0, 2.17332}}], Line[{{0, 2.5}, {y, 2.5}}], Line[{{0, 2.17332}, {y0, 2.17332}}]}}];\)\), "\n", \(\(p10 = Show[p1, p2, p3, p4, p5, p6, p7, p8, p9, PlotRange \[Rule] {{\(-1.5\), 10}, {\(-6\), 4}}, Axes \[Rule] None, AspectRatio \[Rule] 0.7, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p11 = Graphics[{{AbsoluteThickness[0.5], Line[{{\(-1\), \(-5\)}, {6.5, \(-5\)}}], Line[{{0, \(-5.5\)}, {0, \(-1.5\)}}]}, {AbsoluteThickness[ 0.75], Line[{{f[t0, 4], \(-3.75\)}, {f[t1, 7], \(-2.25\)}, {f[ t3, 7], \(-2.25\)}, {f[t2, 4], \(-3.75\)}, {f[t0, 4], \(-3.75\)}}]}, {AbsoluteThickness[2], Line[{{f[t0, 4], c0}, {f[t2, 4], c0}}], Line[{{f[t0, 4], c}, {f[t2, 4], c}}]}, {AbsoluteThickness[ 0.75], Dashing[{0.01, 0.01}], Line[{{0, c0}, {f[t0, 4], c0}}], Line[{{0, c}, {f[t0, 4], c}}], Line[{{y0, \(-5\)}, {y0, c0}}], Line[{{y, \(-5\)}, {y, c}}]}}];\)\), "\n", \(\(p12 = Graphics[{{AbsoluteThickness[0.5], Line[{{8, \(-1.5\)}, {8, \(-5.5\)}}]}, {AbsoluteThickness[2], Line[{{8, \(-3.75\)}, {8, \(-2.25\)}}]}, {AbsolutePointSize[4], Point[{8, \(-5\)}]}, {AbsolutePointSize[5], Point[{8, c0}], Point[{8, c}]}}];\)\), "\n", \(\(p13 = Graphics[{{AbsoluteThickness[0.75], Line[{{5.5, \((c + c0)\)/2}, {7, \((c + c0)\)/2}}], Line[{{y, \(-1.8\)}, {y, \(-0.8\)}}], Line[{{2.52941, \(-1.8\)}, {1.94118, \(-0.8\)}}], Line[{{6, \(-0.56\)}, {7, \(-1.58\)}}]}, arrow[7, \((c + c0)\)/2, 0.075, 0.05, \[Pi]/2], arrow[7, \(-1.58\), 0.075, 0.05, ArcTan[5.1/5] + \[Pi]/2], arrow[y, \(-0.8\), 0.075, 0.05, 0], arrow[1.94118, \(-0.8\), 0.075, 0.05, ArcTan[5.1/3] - \[Pi]/2], Text["\", {0.5, \(-4.75\)}], Text["\", {y0, \(-5.3\)}], Text["\", {y, \(-5.3\)}], Text["\", {7, \(-5\)}], Text["\", {4.5, \(-3.5\)}], Text["\", {y, \(-2.5\)}], Text["\", {\(-0.5\), c0}], Text["\", {\(-0.5\), c}], Text["\", {8.5, \(-5\)}], Text["\", {8.5, \(-3.6\)}], Text["\", {8.5, c0}], Text["\", {8.5, c}], Text["\", {8, \(-1.5\)}], Text["\", {6.8, c0}], Text["\", {6.8, \(-2\)}], Text["\", {5, 1}], Text["\", {3.5, \(-1\)}], Text["\", {1.8, \(-1.5\)}], Text["\", {0.5, 0.25}], Text["\", {y0, \(-0.3\)}], Text["\", {y, \(-0.3\)}], Text["\", {7, 0}], Text["\", {6, 1}], Text["\", {3.45, 2}], Text["\", {2.5, 1.8}], Text["\", {\(-0.5\), 2.2}], Text["\", {\(-0.5\), 2.5}], Text["\", {0, 4}]}];\)\), "\n", \(\(Show[p10, p11, p12, p13, PlotRange \[Rule] {{\(-1.5\), 10}, {\(-6\), 4.5}}, Axes \[Rule] None, AspectRatio \[Rule] 1, Background \[Rule] lightblue, ImageSize \[Rule] 900, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 4.6.3 (Torus) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\n", \(Off[Eliminate::ifun]\), "\n", \(<< Graphics`ContourPlot3D`\), "\n", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\n", \(sol = Eliminate[{x\_1 == \((2 + Cos[\[Theta]])\) Cos[\[Alpha]], x\_2 == \((2 + Cos[\[Theta]])\) Sin[\[Alpha]], x\_3 == Sin[\[Theta]]}, {\[Alpha], \[Theta]}]\), "\n", \(Simplify[sol[\([1]\)]]\), "\n", \(Expand[sol[\([2]\)]]\), "\n", \(g[x1_, x2_, x3_] := \((x1\^2 + x2\^2 + x3\^2 - 5)\)\^2 + 16 \((x3\^2 - 1)\)\), "\n", \(Expand[g[x\_1, x\_2, x\_3]]\), "\n", \(Solve[{\[PartialD]\_\(x\_1\)\ g[x\_1, x\_2, x\_3] \[Equal] 0, \[PartialD]\_\(x\_2\)\ g[x\_1, x\_2, x\_3] \[Equal] 0, \[PartialD]\_\(x\_3\)\ g[x\_1, x\_2, x\_3] \[Equal] 0, g[x\_1, x\_2, x\_3] \[Equal] 0}, {x\_1, x\_2, x\_3}] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]", \(\(ContourPlot3D[ g[x1, x2, x3], {x1, \(-3\), 3}, {x2, \(-3\), 3}, {x3, \(-3\), 3}, PlotPoints \[Rule] {9, 12}, ContourStyle \[Rule] RGBColor[0.7, 0.7, 1], LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[0, 0, 1]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, Boxed \[Rule] False, ImageSize \[Rule] 1000];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Section 4.8 Morse's Lemma", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(r1 = \(-1\);\)\), "\[IndentingNewLine]", \(\(r2 = 0;\)\), "\[IndentingNewLine]", \(\(r3 = 1;\)\), "\[IndentingNewLine]", \(f[ri_, r_, \[Alpha]_] := {\@\(ri + r\^2\)\ Sin[\[Alpha]], \@\(ri + r\^2\)\ \ Cos[\[Alpha]], r}\), "\n", \(\(ParametricPlot3D[ f[r1, r, \[Alpha]], {\[Alpha], 0, 2\ \[Pi]}, {r, \(-3\), r1}, PlotPoints \[Rule] {71, 51}, Boxed \[Rule] False, Axes \[Rule] False, AspectRatio \[Rule] 1.4, ViewPoint \[Rule] {4, \(-1\), 1}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(ParametricPlot3D[ f[r1, r, \[Alpha]], {\[Alpha], 0, 2\ \[Pi]}, {r, \(-r1\), 3}, PlotPoints \[Rule] {71, 51}, Boxed \[Rule] False, Axes \[Rule] False, AspectRatio \[Rule] 1.4, ViewPoint \[Rule] {4, \(-1\), 1}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p1 = Show[%, %%, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(ParametricPlot3D[ f[r2, r, \[Alpha]], {\[Alpha], 0, 2\ \[Pi]}, {r, \(-3\), r2}, PlotPoints \[Rule] {71, 51}, Boxed \[Rule] False, Axes \[Rule] False, AspectRatio \[Rule] 1.4, ViewPoint \[Rule] {4, \(-1\), 1}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(ParametricPlot3D[ f[r2, r, \[Alpha]], {\[Alpha], 0, 2\ \[Pi]}, {r, \(-r2\), 3}, PlotPoints \[Rule] {71, 51}, Boxed \[Rule] False, Axes \[Rule] False, AspectRatio \[Rule] 1.4, ViewPoint \[Rule] {4, \(-1\), 1}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = Show[%, %%, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = ParametricPlot3D[ f[r3, r, \[Alpha]], {\[Alpha], 0, 2\ \[Pi]}, {r, \(-3\), 3}, PlotPoints \[Rule] {71, 51}, Boxed \[Rule] False, Axes \[Rule] False, AspectRatio \[Rule] 1.4, ViewPoint \[Rule] {4, \(-1\), 1}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[GraphicsArray[{p1, p2, p3}], ImageSize \[Rule] 1200, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Morse's Lemma Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Missing. P38SB", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 5: THEORY", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["\<\ Definition 5.1.1 Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Missing. P411D", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 5.3.3 Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Missing. P423V", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 5.3.4 Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Missing. P422V", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 5.3.4 New Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(\(\(sphere[{x1_, x2_, x3_}]\)[\[Alpha]_, \[Theta]_] := {x1 + Cos[\[Alpha]] Cos[\[Theta]], x2 + Sin[\[Alpha]] Cos[\[Theta]], x3 + Sin[\[Theta]]}\)\(\[IndentingNewLine]\) \) (*\ unit\ sphere\ centered\ at\ origin\ *) \ \), "\[IndentingNewLine]", \(\(\(s1 = ParametricPlot3D[ Evaluate[ Append[\(sphere[{0, 0, 0}]\)[\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 0]]]], {\[Alpha], 0, \[Pi]/2}, {\[Theta], \[Pi]/8, \[Pi]/4.5}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] 35, DisplayFunction \[Rule] Identity];\)\(\[IndentingNewLine]\) \) (*\ unit\ sphere\ centered\ at\ \((0, 0, 1)\)\ *) \ \), "\[IndentingNewLine]", \(\(\(s2 = ParametricPlot3D[ Evaluate[ Append[\(sphere[{0, 0, 1}]\)[\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]]]], {\[Alpha], 0, \[Pi]/2}, {\[Theta], \(-\[Pi]\)/5, \(-\[Pi]\)/8}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] 35, DisplayFunction \[Rule] Identity];\)\(\[IndentingNewLine]\) \) (*\ point\ in\ intersection\ of\ two\ spheres\ and\ gradient\ vector\ for\ \ sphere\ s1\ *) \), "\[IndentingNewLine]", \(\(p1 = \(sphere[{0, 0, 0}]\)[\[Pi]/4, \[Pi]/ 6];\)\), "\[IndentingNewLine]", \(\(\(p1 == \(sphere[{0, 0, 1}]\)[\[Pi]/4, \(-\[Pi]\)/ 6]\)\(\[IndentingNewLine]\) \) (*\ gradient\ vector\ for\ sphere\ s2\ *) \ \), "\[IndentingNewLine]", \(\(\(p2 = \(sphere[{0, 0, 1}]\)[\[Pi]/4, \(-\[Pi]\)/6] - {0, 0, 1};\)\(\[IndentingNewLine]\) \) (*\ cross\ product\ of\ p1\ and\ p2\ *) \ \), "\[IndentingNewLine]", \(\(p3 = Cross[p1, p2];\)\), "\[IndentingNewLine]", \(\(\(l1 = Graphics3D[{Thickness[0.003], RGBColor[0, 0.5, 0], Line[{p1, 1.3 p1}]}];\)\(\[IndentingNewLine]\) \) (*\ gradient\ vector\ for\ sphere\ s2\ *) \ \), "\[IndentingNewLine]", \(\(\(l2 = Graphics3D[{Thickness[0.003], darkblue, Line[{p1, p1 + 0.25 p2}]}];\)\(\[IndentingNewLine]\) \) (*\ tangent\ line\ to\ intersection\ of\ the\ spheres\ *) \ \), "\ \[IndentingNewLine]", \(\(\(l3 = Graphics3D[{Thickness[0.003], RGBColor[1, 0, 0], Line[{p1 - p3, p1 + p3}]}];\)\(\[IndentingNewLine]\) \) (*\ intersection\ of\ two\ spheres\ *) \), "\[IndentingNewLine]", \(\(t = ParametricPlot3D[ Append[\(sphere[{0, 0, 0}]\)[\[Alpha], \[Pi]/6], RGBColor[1, 0, 0]], {\[Alpha], 0, \[Pi]/2}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(\(Show[s1, l1, s2, l2, l3, t, BoxRatios \[Rule] {1, 1, 1/3}, ViewPoint \[Rule] {\(-2\), 5, 0.6}, Background \[Rule] lightblue, Axes \[Rule] False, Boxed \[Rule] False, Ticks \[Rule] None, ImageSize \[Rule] 1200, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 5.3.5 Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Point of arrow has been added by means of Xfig.", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\n", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(x10\ \ \ \ = 0.5;\)\), "\n", \(\(x1min = \(-0.3\);\)\), "\n", \(\(x1max = 0.3;\)\), "\n", \(\(x20\ \ \ \ \ = 0.5;\)\), "\n", \(\(x2min\ = \(-0.3\);\)\), "\n", \(\(x2max\ = 0.3;\)\), "\n", \(\(tmin\ \ \ = 0;\)\), "\n", \(\(\(tmax\ \ \ = 1;\)\(\[IndentingNewLine]\) \)\), "\n", \(\(f[x1_, x2_] := \@\(1 - \((x1\^4 + x2\^4)\)\);\)\), "\n", \(\(graph[x1_, x2_] := {x1, x2, f[x1, x2]};\)\), "\[IndentingNewLine]", \(\(h[x1_, x2_, x3_] := x1\^4 + x2\^4 + x3\^2 - 1;\)\), "\n", \(\(\(\(tangentplane[f_]\)[x10_, x20_]\)[h1_, h2_] := f[x10, x20] + \[PartialD]\_xx1\ f[xx1, xx2] h1 + \[PartialD]\_xx2\ f[xx1, xx2] h2 /. {xx1 \[Rule] x10, xx2 \[Rule] x20};\)\), "\[IndentingNewLine]", \(\(grad = {4\ x1\^3, 4\ x2\^3, 2\ x3} /. {x1 \[Rule] x10, x2 \[Rule] x20, x3 \[Rule] f[x10, x20]};\)\), "\n", \(\(\(normal[t_] := graph[x10, x20] + t\ grad;\)\(\n\) \)\), "\[IndentingNewLine]", \(\(g = ParametricPlot3D[ graph[x10 + x1, x20 + x2] // Evaluate, {x1, x1min, x1max}, {x2, x2min, x2max}, Ticks \[Rule] False, Boxed \[Rule] False, Axes \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(t = ParametricPlot3D[\(\(tangentplane[graph]\)[x10, x20]\)[h1, h2] // Evaluate, {h1, x1min, x1max}, {h2, x2min, x2max}, Ticks \[Rule] False, Boxed \[Rule] False, Axes \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(n = ParametricPlot3D[normal[t], {t, tmin, tmax}, Ticks \[Rule] False, Boxed \[Rule] False, Axes \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(l = Graphics3D[{Thickness[0.004], RGBColor[0, 0.5, 0], Line[{{1. , 1. , 2.806243040080456}, {0.5, 0.5, 0.9354143466934853}}]}];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(Show[g, t, l, ViewPoint \[Rule] {1.2, 2, \(-0.4\)}, ImageSize \[Rule] 900, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 5.3.6 (Cycloid) Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(darkblue = RGBColor[34/256, 44/256, 162/256]; p1 = ParametricPlot[{t - Sin[t], 1 - Cos[t]}, {t, \(-1.5\), 2\ \[Pi] + 1.5}, PlotStyle \[Rule] RGBColor[1, 0, 0], AspectRatio \[Rule] Automatic, PlotStyle \[Rule] AbsoluteThickness[1.5], DisplayFunction \[Rule] Identity];\), "\n", \(\(p2 = Graphics[{AbsoluteThickness[1], darkblue, Dashing[{0.0125, 0.0125}], Circle[{1.5, 1}, 1], Circle[{2\ \[Pi] - 1.5, 1}, 1]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = Graphics[{AbsolutePointSize[5], Point[{1.5 - Sin[1.5], 1 - Cos[1.5]}], Point[{2\ \[Pi] - 1.5 - Sin[2\ \[Pi] - 1.5], 1 - Cos[2\ \[Pi] - 1.5]}]}, PlotRange \[Rule] {0, 3}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p4 = Show[p1, p2, p3, AspectRatio \[Rule] 1, Ticks \[Rule] {{0, \[Pi], 2\ \[Pi]}, {1}}];\)\), "\n", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-4\)\ h} + {x, y}, {x, y}, R[a] . {2\ b, \(-4\)\ h} + {x, y}, R[a] . {0, \(-3\)\ h} + {x, y}}]\), "\[IndentingNewLine]", \(\(\(Show[p4, Graphics[{darkblue, arrow[\[Pi], 2, 0.075, 0.05, 0.49\ \[Pi]], arrow[0.7\ \[Pi] - Sin[0.7\ \[Pi]], 1 - Cos[0.7\ \[Pi]], 0.075, 0.05, 0.325\ \[Pi]], arrow[1.4\ \[Pi] - Sin[1.4\ \[Pi]], 1 - Cos[1.4 \[Pi]], 0.075, 0.05, 0.695\ \[Pi]]}], AxesStyle \[Rule] RGBColor[0, 0.5, 0], AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, ImageSize \[Rule] 900, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\[Phi][t_] := {t - Sin[t], 1 - Cos[t]}\), "\[IndentingNewLine]", \(\(\[Phi]'\)[t]\), "\[IndentingNewLine]", \(\[Nu][t_] := FullSimplify[Norm[\(\[Phi]'\)[t]], Im[t] \[Equal] 0]\), "\[IndentingNewLine]", \(\[Nu][t]\), "\[IndentingNewLine]", \(Print["\"]\), \ "\[IndentingNewLine]", \(\(Plot[\[Nu][t], {t, \(-1.5\), 2 \[Pi] + 1.5}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, PlotRange \[Rule] {{\(-1\), 2 \[Pi] + 1}, {\(-0.2\), 2.2}}, AspectRatio \[Rule] Automatic, ImageSize \[Rule] 900];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Tau][t_] := 1\/\[Nu][t]\ \(\[Phi]'\)[t]\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[\(\[Phi]'\)[t], {t, 0, 2 \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotRange \[Rule] {{\(-0.1\), 2.2}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[\[Tau][t], {t, 0, 2 \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotRange \[Rule] {{\(-0.1\), 1.2}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[GraphicsArray[{p1, p2}], Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 5.3.7 (Descartes' folium) Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\[IndentingNewLine]", \(a = 1;\), "\[IndentingNewLine]", \(\[Phi][ t_] := \(\(3 a\)\/\(1 + t\^3\)\) {t, t\^2}\), "\[IndentingNewLine]", \(f1[ t_] := \(\[Phi][t]\)[\([1]\)]\), "\n", \(f2[ t_] := \(\[Phi][t]\)[\([2]\)]\), "\n", \(R[ a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-4\)\ h} + {x, y}, {x, y}, R[a] . {2\ b, \(-4\)\ h} + {x, y}, R[a] . {0, \(-3\)\ h} + {x, y}}]\), "\n", \(p1 = ParametricPlot[\[Phi][t], {t, \(-0.8\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], PlotPoints \[Rule] 30, PlotStyle \[Rule] {AbsoluteThickness[1]}, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] Identity];\), "\n", \(p2 = ParametricPlot[RotateLeft[\[Phi][t]], {t, \(-0.8\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], PlotPoints \[Rule] 30, PlotStyle \[Rule] {AbsoluteThickness[1]}, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] Identity];\), "\n", \(p3 = Graphics[{AbsoluteThickness[0.5], Dashing[{0.025, 0.025}], darkblue, Line[{{\(-3\), 2}, {2, \(-3\)}}]}];\), "\n", \(p4 = Graphics[{darkblue, arrow[f1[\(-0.4\)], f2[\(-0.4\)], 0.05, 0.04, 0.71\ \[Pi]], arrow[f1[2], f2[2], 0.05, 0.04, 1.3\ \[Pi]], arrow[f1[\(-2.2\)], f2[\(-2.2\)], 0.05, 0.04, 2.8\ \[Pi]], {AbsoluteThickness[0.5], Line[{{0, 0}, {\(-0.15\), 0.15}}], Line[{{0, 0}, {0.15, \(-0.15\)}}]}, Text["\<-a\>", {\(-1\), 0}, {1.5, 2}], Text["\<-a\>", {0, \(-1\)}, {2, 1}], {AbsolutePointSize[5], Point[{0, 0}], Point[{f1[\(-0.5\)], f2[\(-0.5\)]}], Point[{f1[\(-2\)], f2[\(-2\)]}], Point[{f1[1], f2[1]}]}, Text["\", {f1[1], f2[1]}, {\(-2\), 0}], Text["\", \[Phi][\(-2\)], {\(-2\), \(-0.5\)}], Text["\", \[Phi][\(-0.5\)], {\(-1.5\), \(-0.5\)}], Text["\", {0, 0}, {2, \(-5\)}], Text["\", {0, 0}, {\(-3\), 3.5}]}];\), "\n", RowBox[{\(Show[p1, p2, p3, p4, PlotRange \[Rule] {{\(-3.5\), 3.5}, {\(-2.5\), 2}}, Ticks \[Rule] None, DefaultFont \[Rule] 12. , AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 900];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print[\*"\"\\""]\), "\n", \(\[Phi][ t_] := \(\(3 a\)\/\(1 + t\^3\)\) {t, t\^2}\), "\n", \(p5 = ParametricPlot[Evaluate[\[Phi][t]], {t, \(-1\)/2, 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-2\), 2}, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\), "\n", \(p6 = ParametricPlot[Evaluate[\[Phi][1/t]], {t, \(-1\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-0.2\), 1.7}, {\(-2\), 2}}, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", RowBox[{\(Show[GraphicsArray[{p5, p6}], Background \[Rule] lightblue, ImageSize \[Rule] 900, DisplayFunction \[Rule] $DisplayFunction];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(Off[ General::spell1]\), "\[IndentingNewLine]", \(g[x1_, x2_] := x1\^3 + x2\^3 - 3 a\ x1\ x2\), "\[IndentingNewLine]", \(gradg[{x1_, x2_}] := {\[PartialD]\_xx1\ g[xx1, xx2], \[PartialD]\_xx2\ g[xx1, xx2]} /. {xx1 \[Rule] x1, xx2 \[Rule] x2}\), "\[IndentingNewLine]", \(grad[t_] := 0.1 Simplify[\(Composition[gradg, \[Phi]]\)[ t]]\), "\[IndentingNewLine]", \(normal[t_] := {darkblue, Line[{\[Phi][t], \[Phi][t] + grad[t]}]}\), "\[IndentingNewLine]", \(grads1 = Graphics[Evaluate[Table[normal[t], {t, \(-1\)/2, 1, 0.005}]], DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(curve1 = ParametricPlot[Evaluate[\[Phi][t]], {t, \(-1\)/2, 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-2\), 2}, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(p7 = Show[curve1, grads1, AspectRatio \[Rule] Automatic, Axes \[Rule] None, PlotRange \[Rule] All, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(grads2 = Graphics[Evaluate[Table[normal[1/t], {t, \(-1\)/2, 1, 0.005}]], DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(curve2 = ParametricPlot[Evaluate[\[Phi][1/t]], {t, \(-1\)/2, 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-2\), 2}, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(p8 = Show[curve2, grads2, AspectRatio \[Rule] Automatic, Axes \[Rule] None, PlotRange \[Rule] All, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(Show[ p7, p8, ImageSize \[Rule] 900, DisplayFunction -> $DisplayFunction];\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\)\>\"", "]"}], "\[IndentingNewLine]", \(a = 1;\), "\[IndentingNewLine]", \(p9 = ContourPlot[g[x1, x2], {x1, \(-4\), 4}, {x2, \(-4\), 4}, \ Contours \[Rule] 251, PlotPoints \[Rule] 300, \ ColorFunction \[Rule] Hue, Ticks \[Rule] None, \ Frame \[Rule] False, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(Show[ p9, ImageSize \[Rule] 915, DisplayFunction \[Rule] $DisplayFunction];\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"to\",\nFontVariations->{\"CompatibilityType\"->0}]\)\!\(\* StyleBox[\" \",\nFontVariations->{\"CompatibilityType\"->0}]\)\!\(\* StyleBox[\"bisectrix\",\nFontVariations->{\"CompatibilityType\"->0}]\)\!\(\* StyleBox[\" \",\nFontVariations->{\"CompatibilityType\"->0}]\)\!\(\* StyleBox[\"of\",\nFontVariations->{\"CompatibilityType\"->0}]\)\!\(\* StyleBox[\" \",\nFontVariations->{\"CompatibilityType\"->0}]\)\!\(\* StyleBox[\"plane\",\nFontVariations->{\"CompatibilityType\"->0}]\)\>\"", "]"}], "\[IndentingNewLine]", \(Plot[g[t, t], {t, \(-0.5\), 1.7}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, ImageSize \[Rule] 400];\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 5.3.8 (Cusp of a plane curve) New Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\[Gamma][t_] := {t\^2, t\^3}\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(ParametricPlot[\[Gamma][t], {t, \(-1\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-0.1\), 1.1}, {\(-1.1\), 1.1}}, ImageSize \[Rule] 300, Ticks \[Rule] None];\)\), "\[IndentingNewLine]", \(Eliminate[{x\_1, x\_2} == \[Gamma][t], t]\), "\[IndentingNewLine]", \(\(\[Gamma]'\)[t]\), "\[IndentingNewLine]", \(\(Solve[\(\[Gamma]'\)[t] == {0, 0}, t]\)[\([1]\)] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]", \(\[Nu][t_] := FullSimplify[Norm[\(\[Gamma]'\)[t]], Im[t] \[Equal] 0]\), "\[IndentingNewLine]", \(\[Nu][t]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(T[t_] := 1\/\[Nu][t]\ \(\[Gamma]'\)[t]\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[T[t], {t, \(-1\), 0}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotRange \[Rule] {{\(-1\), 0}, {0, 1}}, AspectRatio \[Rule] Automatic, ImageSize \[Rule] 300, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = Graphics[{darkblue, AbsolutePointSize[5], Point[T[\(-1\)]]}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = Show[p1, p2];\)\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot[T[t], {t, 0, 1}, Descartes'\ folium\ with\ vectors\ \((uniformly)\)\ proportional\ \ to\ the\ gradient\ vectorsPlotRange \[Rule] {{0, 1}, {0, 1}}, AspectRatio \[Rule] Automatic, ImageSize \[Rule] 300, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p5 = Graphics[{darkblue, AbsolutePointSize[5], Point[T[1]]}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p6 = Show[p4, p5];\)\), "\[IndentingNewLine]", \(\(Show[GraphicsArray[{p3, p6}], Background \[Rule] lightblue, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\)\), \ "\[IndentingNewLine]", \(\(\[Gamma]''\)[0]\), "\[IndentingNewLine]", \(\(\[Gamma]'''\)[0]\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(\[Delta][t_] := {t\^2, t\^7}\), "\[IndentingNewLine]", \(\(ParametricPlot[\[Delta][t], {t, \(-1\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-0.1\), 1.1}, {\(-1.1\), 1.1}}, ImageSize \[Rule] 300, Ticks \[Rule] None];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Section 5.7 Gaussian curvature of surface", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\n", \(f[t_] := 5\ Sin[t/4] + 4\), "\n", \(\(Plot[f[t], {t, 0, 3\ \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], PlotRange \[Rule] {{\(-2\), 11}, {\(-2\), 11}}, AspectRatio \[Rule] 1, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{darkblue, Line[{{2 \[Pi], f[2 \[Pi]]}, {2 \[Pi], f[2 \[Pi]] + 2}}], Line[{{0, f[0]}, {\(-10\)/Sqrt[41], 8 + 8/Sqrt[41] - f[0]}}], Line[{{3\ \[Pi], f[3\ \[Pi]]}, {3\ \[Pi] + 10/Sqrt[57], f[3\ \[Pi]] + 8\ Sqrt[2/57]}}]}];\)\), "\n", \(\(Graphics[{darkblue, Line[{{\[Pi], 0}, {3\ \[Pi], 0}}], Line[{{2\ \[Pi], 0}, {2\ \[Pi], 2}}], Line[{{2\ \[Pi], 0}, {2\ \[Pi] - 10/Sqrt[41], 8/Sqrt[41]}}], Line[{{2 \[Pi], 0}, {2 \[Pi] + 10/Sqrt[57], 8\ Sqrt[2/57]}}], {RGBColor[1, 0, 0], Circle[{2\ \[Pi], 0}, 2, {0.27\ \[Pi], 0.785\ \[Pi]}]}}];\)\), "\n", \(\(Graphics[{Text["\", {2 \[Pi], f[2 \[Pi]]}, {0, 3}], Text["\", {1.5, 5}], Text["\<0\>", {2\ \[Pi], 0}, {0, 2}], Text["\", {6.5, 2.5}], Text["\", {2\ \[Pi] - 1.5, 1.75}], Text["\", {2\ \[Pi] + 2, 0}, {\(-1\), \(-3\)}]}];\)\), "\n", \(\(Show[%%%%, %%%, %%, %, Axes \[Rule] None, DefaultFont \[Rule] 12. , Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Section 5.11 Transversality", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\n", \(f[x_] := 0.2\ x\^2\), "\n", \(g[n_, x_] := \(-\(f[x]\/n\)\)\), "\n", \(\(a = 0.7;\)\), "\n", \(\(\(h[x_] := 2\ \@x\)\(\[IndentingNewLine]\) \)\), "\n", \(\(Plot[\(-f[x]\), {x, \(-1\), 1}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[{2\ Sqrt[x], \(-2\)\ Sqrt[x]}, {x, 0, 0.5}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(fig1 = Show[%%, %, Graphics[Text["\", {0, \(-1.45\)}]], DefaultColor \[Rule] darkblue, DefaultFont \[Rule] 12. , AspectRatio \[Rule] 1, Axes \[Rule] False, PlotRange \[Rule] {{\(-1.5\), 1.5}, Automatic}, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(Plot[{f[x + 0.5], \(-f[x + 0.5]\)}, {x, \(-2\), \(-0.5\)}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[0, {x, \(-0.5\), 0.5}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[{f[x - 0.5], \(-f[x - 0.5]\)}, {x, 0.5, 2}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(fig2 = Show[%%%, %%, %, Graphics[Text["\", {0, \(-0.5\)}]], DefaultColor \[Rule] darkblue, DefaultFont \[Rule] 12. , Axes \[Rule] False, PlotRange \[Rule] {{\(-1.5\), 1.5}, Automatic}, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(Plot[{f[x], \(-f[x]\)}, {x, \(-1\), 1}, Axes \[Rule] False, PlotRange \[Rule] {{\(-1.5\), 1.5}, Automatic}, PlotPoints \[Rule] 100, AspectRatio \[Rule] 0.7, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(fig3 = Show[%, Graphics[Text["\", {0, \(-0.25\)}]], DefaultColor \[Rule] darkblue, DefaultFont \[Rule] 12. , Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(Plot[g[3, x] + a, {x, \(-2.5\), 2.5}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] - a, {x, \(-2\), 2}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{Line[{{\(-2.5\), g[3, \(-2.5\)] + a}, {\(-2\), g[3, \(-2\)] - a}}], Line[{{2.5, g[3, 2.5] + a}, {2, g[3, 2] - a}}], {AbsolutePointSize[ 4], Point[{0, 0}]}}];\)\), "\n", \(\(Plot[2\ \@x, {x, 0, 0.5}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-2\)\ \@x, {x, 0, 0.122852}, PlotStyle \[Rule] Dashing[{0.01, 0.01}], PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-2\)\ \@x, {x, 0.122852, 0.5}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(fig4 = Show[%%%%%%, %%%%%, %%%%, %%%, %%, %, Graphics[Text["\", {0, \(-1.5\)}]], DefaultColor \[Rule] darkblue, DefaultFont \[Rule] 12. , AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-3\), 3}, Automatic}, Axes \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\(\n\) \)\), "\n", \(\(Plot[g[3, x] + a, {x, \(-2.5\), 2.5}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] - a, {x, \(-2\), 2}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{Line[{{\(-2.5\), g[3, \(-2.5\)] + a}, {\(-2\), g[3, \(-2\)] - a}}], Line[{{2.5, g[3, 2.5] + a}, {2, g[3, 2] - a}}], {AbsolutePointSize[ 4], Point[{0, 0}]}}];\)\), "\n", \(\(Plot[2\ f[x + 0.5], {x, \(-2.5\), \(-0.5\)}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[0, {x, \(-0.5\), 0.5}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-4\)\ f[x - 0.5], {x, 0.5, 1.53507}, PlotStyle \[Rule] {Dashing[{0.01, 0.01}]}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-4\)\ f[x - 0.5], {x, 1.53507, 2.25}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(fig5 = Show[%%%%%%%, %%%%%%, %%%%%, %%%%, %%%, %%, %, Graphics[Text["\", {0, \(-1.5\)}]], DefaultColor \[Rule] darkblue, DefaultFont \[Rule] 12. , AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-3\), 3}, Automatic}, Axes \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\(\n\) \)\), "\n", \(\(Plot[g[3, x] + a, {x, \(-2.5\), 2.5}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] - a, {x, \(-2\), 2}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{Line[{{\(-2.5\), g[3, \(-2.5\)] + a}, {\(-2\), g[3, \(-2\)] - a}}], Line[{{2.5, g[3, 2.5] + a}, {2, g[3, 2] - a}}], {AbsolutePointSize[ 4], Point[{0, 0}]}}];\)\), "\n", \(\(Plot[1.2\ x\^2, {x, \(-1\), 1}, PlotPoints \[Rule] 100, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(fig6 = Show[%%%%, %%%, %%, %, Graphics[Text["\", {0, \(-1.5\)}]], DefaultColor \[Rule] darkblue, DefaultFont \[Rule] 12. , AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-3\), 3}, Automatic}, Axes \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(\(Show[ GraphicsArray[{{fig1, fig2, fig3}, {fig4, fig5, fig6}}]];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(b = 0.85;\)\), "\n", \(\(b1 = 0.5;\)\), "\n", \(\(b2 = 0.65;\)\), "\[IndentingNewLine]", \(\(a = 0.5;\)\), "\n", \(\(Plot[g[3, x] + a, {x, \(-2.5\), \(-0.440695\)}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] + a, {x, \(-0.440695\), 0.310899}, PlotStyle \[Rule] {Dashing[{0.01, 0.01}]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] + a, {x, 0.310899, 2.5}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] - a, {x, \(-2\), 2}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[h[x + 0.5], {x, \(-0.5\), \(-0.440695\)}, PlotStyle \[Rule] {Dashing[{0.01, 0.01}]}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[h[x + 0.5], {x, \(-0.440695\), 0.1}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-h[x + 0.5]\), {x, \(-0.5\), \(-0.440695\)}, PlotStyle \[Rule] {Dashing[{0.01, 0.01}]}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-h[x + 0.5]\), {x, \(-0.440695\), 0.1}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[{h[x - 0.25], \(-h[x - 0.25]\)}, {x, 0.25, 0.8}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{Line[{{\(-2.5\), g[3, \(-2.5\)] + a}, {\(-2\), g[3, \(-2\)] - a}}], Line[{{2.5, g[3, 2.5] + a}, {2, g[3, 2] - a}}], Line[{{0.1, h[0.6]}, {0.8, h[0.55]}}], Line[{{0.1, \(-h[0.6]\)}, {0.8, \(-h[0.55]\)}}], Line[{{\(-0.440695\), h[\(-0.440695\) + 0.5]}, {0.314156, \(-h[ 0.314156 - 0.25]\)}}]}];\)\), "\n", \(\(\(Show[%%%%%%%%%%, %%%%%%%%%, %%%%%%%%, %%%%%%%, %%%%%%, %%%%%, %%%%, \ %%%, %%, %, Graphics[Text["\", {0, \(-2\)}]], DefaultFont \[Rule] 12. , DefaultColor \[Rule] darkblue, Axes \[Rule] None, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-3\), 3}, Automatic}, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(a = 0.5;\)\), "\n", \(\(Plot[g[3, x] + a, {x, \(-2.5\), \(-2.01165\)}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] + a, {x, \(-2.01165\), 1.62204}, PlotStyle \[Rule] {Dashing[{0.01, 0.01}]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] + a, {x, 1.62204, 2.5}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] - a, {x, \(-2\), 2}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-g[b, x + 0.25]\) - a, {x, \(-2.5\), 2.5}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-g[b2, x]\) + a, {x, \(-2\), 0}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[\(-g[b1, x]\) + a, {x, 0, 2}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{Line[{{\(-2.5\), g[3, \(-2.5\)] + a}, {\(-2\), g[3, \(-2\)] - a}}], Line[{{2.5, g[3, 2.5] + a}, {2, g[3, 2] - a}}], Line[{{\(-2.5\), \(-g[b, \(-2.25\)]\) - a}, {\(-2\), \(-g[b2, \(-2\)]\) + a}}], Line[{{2.5, \(-g[b, 2.75]\) - a}, {2, \(-g[b1, 2]\) + a}}], Line[{{0, a}, {\(-0.25\), \(-g[b, 0]\) - a}}]}];\)\), "\n", \(\(\(Show[%%%%%%%%, %%%%%%%, %%%%%%, %%%%%, %%%%, %%%, %%, %, Graphics[Text["\", {0, \(-1.5\)}]], DefaultFont \[Rule] 12. , DefaultColor \[Rule] darkblue, Axes \[Rule] None, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-3\), 3}, Automatic}, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(a = 0.5;\)\), "\n", \(\(Clear[r];\)\), "\n", \(\(Clear[s];\)\), "\n", \(r[t_] := 1\), "\n", \(s[t_] := 0.2\), "\n", \(\(Plot[g[3, x] + a, {x, \(-2.5\), \(-0.628281\)}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] + a, {x, 0.628281, 2.5}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[g[3, x] - a, {x, \(-2\), 2}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{Line[{{\(-2.5\), g[3, \(-2.5\)] + a}, {\(-2\), g[3, \(-2\)] - a}}], Line[{{2.5, g[3, 2.5] + a}, {2, g[3, 2] - a}}], {AbsolutePointSize[ 4], Point[{0, 0}]}}];\)\), "\n", \(\(Plot[1.2\ x^2, {x, \(-1\), 1}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(ParametricPlot[{r[t]\ Cos[t], s[t]\ Sin[t] + 1.2}, {t, 0, 2\ \[Pi]}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[%%%%%%, %%%%%, %%%%, %%%, %%, %, Graphics[Text["\", {0, \(-1.5\)}]], DefaultFont \[Rule] 12. , DefaultColor \[Rule] darkblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-3\), 3}, Automatic}, Axes \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 0: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 0.1 (Rational parametrization of circle)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica ", FontSlant->"Italic"], "needs some assistance in the exercise. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Solve::ifun]\), "\[IndentingNewLine]", \(tt = Sin[\[Alpha]]\/\(1 + Cos[\[Alpha]]\)\), "\[IndentingNewLine]", \(Simplify[ tt == \(1 - Cos[\[Alpha]]\)\/Sin[\[Alpha]]]\), "\[IndentingNewLine]", \(Solve[t == \(1 - Cos[\[Alpha]]\)\/Sin[\[Alpha]], Cos[\[Alpha]]]\), "\[IndentingNewLine]", \(sol = Flatten[Solve[{t == \(1 - \ Cos[\[Alpha]\[Alpha]]\)\/Sin[\[Alpha]\[Alpha]], Cos[\[Alpha]\[Alpha]] \[Equal] \@\(1 - \ Sin[\[Alpha]\[Alpha]]^2\)}, \[Alpha]\[Alpha]]]\ \), "\[IndentingNewLine]", \(\[Alpha] = \[Alpha]\[Alpha] /. sol[\([2]\)]\), "\[IndentingNewLine]", \(Cos[\[Alpha]]\), "\[IndentingNewLine]", \(\(\(Simplify[Sin[\[Alpha]], t > 0]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Clear[\[Alpha], t]\), "\[IndentingNewLine]", \(\(\[Alpha] = ArcTan[t];\)\), "\[IndentingNewLine]", \(Cos[\[Alpha]]\^2\), "\[IndentingNewLine]", \(Sin[\[Alpha]]\^2\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 0.3 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " does not produce the desired formula directly. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(FullSimplify[ArcTan[1\/x] + ArcTan[x], x > 0]\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_x %]\), "\[IndentingNewLine]", \(%% /. x \[Rule] 1\), "\[IndentingNewLine]", \(%%% /. x \[Rule] \(-1\)\), "\[IndentingNewLine]", \(Limit[x \((\[Pi]\/2 - ArcTan[x])\), x \[Rule] \[Infinity]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 0.6 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(f[x_] := 1\/\@\(\((x - a)\) \((b - x)\)\)\), "\[IndentingNewLine]", \(FullSimplify[\[Integral]f[x] \[DifferentialD]x, {a < x < b}]\), "\[IndentingNewLine]", \(FullSimplify[\[Integral]f[x] \[DifferentialD]x \[Equal] \(-2\) ArcTan[\@\(\(b - x\)\/\(x - a\)\)], {Im[a] \[Equal] 0, Im[b] \[Equal] 0, a < x < b}]\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_x\ ArcSin[\(2 x - b - a\)\/\(b - a\)], {a < x < b}]\), "\[IndentingNewLine]", \(Simplify[ Together[\[PartialD]\_x\ ArcCos[\(b + a - 2 x\)\/\(b - a\)]], {a < x < b, b - a > 0}]\), "\[IndentingNewLine]", \(PowerExpand[ Simplify[\(-2\) \[PartialD]\_x ArcTan[\@\(\(b - x\)\/\(x - a\)\)], {a < x < b, b - a > 0}]]\), "\[IndentingNewLine]", \(Integrate[f[x], {x, a, b}, Assumptions \[Rule] {a < x < b}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 0.7 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is unable to directly verify all desired identities." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(i[\[Alpha]_] = FullSimplify[Integrate[1\/Cos[\[Alpha]], \[Alpha]], Sin[\[Alpha]] \[Equal] 2 Cos[\[Alpha]\/2]\^2 - 1]\), "\[IndentingNewLine]", \(j[\[Alpha]_] = \(1\/2\) Log[\(1 + Sin[\[Alpha]]\)\/\(1 - Sin[\[Alpha]]\)]\), "\ \[IndentingNewLine]", \(FullSimplify[ i[\[Alpha]] - j[\[Alpha]], {Sin[\[Alpha]] == 2 Sin[\[Alpha]\/2] Cos[\[Alpha]\/2], 0 < \[Alpha] < \[Pi]\/2}]\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_\[Alpha] j[\[Alpha]]]\), "\[IndentingNewLine]", \(i[0] - j[0]\), "\[IndentingNewLine]", \(\(Plot[i[\[Alpha]] - j[\[Alpha]], {\[Alpha], 0, \[Pi]\/2}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(k[\[Alpha]_] = Log[Tan[\[Alpha]\/2 + \[Pi]\/4]]\), "\[IndentingNewLine]", \(PowerExpand[ FullSimplify[ i[\[Alpha]] - k[\[Alpha]], {Cos[\[Alpha]] \[Equal] 1 - 2 Sin[\[Alpha]\/2]\^2, 0 < \[Alpha] < \[Pi]\/2}]]\), "\[IndentingNewLine]", \(\(\[Alpha] = \[Pi]/3.5;\)\), "\[IndentingNewLine]", \(p[t_] := {Cos[t], Sin[t]}\), "\[IndentingNewLine]", \(\(pl[1] = ParametricPlot[p[t], {t, \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] darkblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(pl[2] = ParametricPlot[0.2 p[t], {t, 0, \[Alpha]}, PlotStyle -> RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(pl[3] = Graphics[{darkblue, PointSize[0.015], Point[0 p[0]], Point[p[\[Pi]/2]], Point[p[\[Alpha]]], Point[{Tan[\[Alpha]/2 + \[Pi]/4], 0}]}];\)\), "\[IndentingNewLine]", \(\(pl[4] = Graphics[{darkblue, Line[{p[\[Alpha]], 0 p[0], p[\[Pi]/2], {Tan[\[Alpha]/2 + \[Pi]/4], 0}, 0 p[0]}]}];\)\), "\[IndentingNewLine]", \(\(Show[Table[pl[j], {j, 4}], Axes \[Rule] False, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 0.8 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(ComplexExpand[ Integrate[\[ExponentialE]\^\(\((\(-p\) + \[ImaginaryI]\ q)\) x\), {x, 0, \[Infinity]}, Assumptions -> {p > 0, Im[q] \[Equal] 0}]]\), "\[IndentingNewLine]", \(Integrate[\(\[ExponentialE]\^\(\(-p\)\ x\)\) Cos[q\ x], {x, 0, \[Infinity]}, Assumptions -> {p > 0, Im[q] \[Equal] 0}]\), "\[IndentingNewLine]", \(Integrate[\(\[ExponentialE]\^\(\(-p\)\ x\)\) Sin[q\ x], {x, 0, \[Infinity]}, Assumptions -> {p > 0, Im[q] \[Equal] 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 0.13 (Partial-fraction decomposition of trigonometric \ functions, and Wallis' product)\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(8 x\ \(\[Sum]\+\(k = 1\)\%\[Infinity] 1\/\(\((2 k - 1)\)\^2 - 4 x\^2\ \)\)\), "\[IndentingNewLine]", \(Simplify[\(-Tan[\[Pi] \((x + 1\/2)\)]\)]\), "\[IndentingNewLine]", \(Simplify[ 1\/x + 2 x\ \(\[Sum]\+\(k = 1\)\%\[Infinity] 1\/\(x\^2 - k\^2\)\)]\), "\ \[IndentingNewLine]", \(Simplify[\[Pi]\ Cot[\[Pi] \(\(\ \)\(x\)\)\/2] - \[Pi]\ Cot[\[Pi] \( x + \ 1\)\/2]]\), "\[IndentingNewLine]", \(Simplify[ 1\/x + 2 x\ \(\[Sum]\+\(k = 1\)\%\[Infinity]\((\(-1\))\)\^k\/\(x\^2 - k\^2\)\ \)]\), "\[IndentingNewLine]", \(Simplify[ 2\ \(\[Sum]\+\(k = 1\)\%\[Infinity]\(\((\(-1\))\)\^k\) \(2 k - \ 1\)\/\(x\^2 - \((2 k - 1)\)\^2\)\)]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[\[Pi]\ Cot[\[Pi]\ x] - 1\/x, x]]\), "\[IndentingNewLine]", \(Simplify[\[Pi]\ x\ \(\[Product]\+\(k = 1\)\%\[Infinity]\((1 - x\^2\/k\^2)\)\), x > 0]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Product]\+\(n = 1\)\%\[Infinity]\((1 - 1\/\(4 n\^2\))\)\), "\[IndentingNewLine]", \(Limit[\((2\^n\ \(n!\))\)\^2\/\(\(\((2 n)\)!\) \@\(2 n + 1\)\), n \[Rule] \[Infinity]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(f[x_] := \[Pi]\/\[Omega]\ Cot[\(\[Pi]\ x\)\/\[Omega]]\), "\ \[IndentingNewLine]", \(Simplify[\(f'\)[x] \[Equal] \(-f[x]\^2\) - 6 \(\[Sum]\+\(k = 1\)\%\[Infinity] 1\/\((k\ \[Omega])\)\^2\)]\)}], \ "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Exercise 0.14 (", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(\[DoubleStruckCapitalR]\_+\)\)]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"sin", " ", StyleBox["x", FontSlant->"Italic"]}], StyleBox["x", FontSlant->"Italic"]], TraditionalForm]]], StyleBox["dx", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ")" }], "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\[Sum]\+\(k = \(-\[Infinity]\)\)\%\[Infinity] Sin[x + k\ \ \[Pi]]\^2\/\((x + k\ \[Pi])\)\^2\), "\[IndentingNewLine]", \(Integrate[ Sin[x]\^2\/x\^2, {x, \(-\[Infinity]\), \[Infinity]}]\), "\ \[IndentingNewLine]", \(Integrate[Sin[x]\/x, {x, 0, \[Infinity]}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 0.15 (Special values of Beta function) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Simplify[\[Sum]\+\(k = 0\)\%\[Infinity]\((\(-1\))\)\^k\ x\^\(p + k - \ 1\), {0 < x < 1}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Simplify[ Integrate[x\^\(p - 1\)\/\(x + 1\), {x, 0, 1}, Assumptions \[Rule] {0 < p < 1}] == \[Sum]\+\(k = 0\)\%\[Infinity]\((\(-1\))\)\^k\/\(p + k\ \)]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[x\^\(p - 1\)\/\(x + 1\), {x, 1, \[Infinity]}, Assumptions \[Rule] {0 < p < 1}] == \[Sum]\+\(k = 1\)\%\[Infinity]\((\(-1\))\)\^k\/\(p - k\ \)]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[x\^\(p - 1\)\/\(x + 1\), {x, 0, \[Infinity]}, Assumptions \[Rule] {0 < p < 1}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[\(\(x\^\(p - 1\)\) Log[x]\)\/\(x - 1\), {x, 0, \[Infinity]}, Assumptions \[Rule] {0 < p < 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 0.16 (Bernoulli polynomials, Bernoulli numbers, and \ power series expansion for tangent) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ "Apparently ", StyleBox["Mathematica", FontSlant->"Italic"], " does neither recognize the generating function relation for the Bernoulli \ polynomials nor the Bernoulli summation formula. \nFor no good reason \ Mathematica adds asymptotes to the illustration, which we have had to remove \ by hand." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Table[BernoulliB[n, x], {n, 0, 5}]\), "\[IndentingNewLine]", \(Factor[%[\([4]\)]]\), "\[IndentingNewLine]", \(Factor[%%[\([6]\)]]\), "\[IndentingNewLine]", \(FullSimplify[\[Sum]\+\(n = 0\)\%\[Infinity]\( BernoulliB[n, x] t\^n\)\/\ \(n!\) == \(t\ \[ExponentialE]\^\(x\ t\)\)\/\(\[ExponentialE]\^t - 1\), t > 0]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ 1 \[Equal] \((\[Sum]\+\(n = 1\)\%\[Infinity] t\^\(n - 1\)\/\(n!\))\) \ \((\[Sum]\+\(n = 0\)\%\[Infinity]\( BernoulliB[n] t\^n\)\/\(n!\))\), t > 0]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Simplify[ t\/\(\[ExponentialE]\^t - 1\) + t\/2 \[Equal] 1 + \[Sum]\+\(n = 2\)\%\[Infinity]\( BernoulliB[n] t\^n\)\/\(n!\), t > 0]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Table[BernoulliB[2 n], {n, 1, 6}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[\((p + 1)\) \(\[Sum]\+\(k = 1\)\%n k\^p\) \[Equal] n\^p + \(n\^\(p + 1\)\) \(\[Sum]\+\(k = 0\)\%p\( Binomial[p + 1, k] \ BernoulliB[k]\)\/n\^k\), {Element[p\ , Integers], p > 0}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[\[Sum]\+\(n = 1\)\%\[Infinity]\(\(\((\(-1\))\)\^\(n - \ 1\)\) \(2\^\(2 n\)\) \((2\^\(2 n\) - 1)\) BernoulliB[2 n] x\^\(2 n - \ 1\)\)\/\(\((2 n)\)!\) \[Equal] Tan[x], {\(-\(\[Pi]\/2\)\) < x < \[Pi]\/2}]\), "\[IndentingNewLine]", \(\(\(Series[Tan[x], {x, 0, 19}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(p1 = Plot[Evaluate[ Normal[Series[Tan[x], {x, 0, 19}]]], {x, \(-3\) \[Pi]/4, 3 \[Pi]/4}, PlotRange \[Rule] {\(-58\), 58}, PlotStyle \[Rule] darkblue, Background \[Rule] lightblue, ImageSize \[Rule] 900, Ticks \[Rule] {{\(-\[Pi]\)/2, \[Pi]/2}, {\(-40\), \(-20\), 20, 40}}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Plot[Evaluate[Tan[x]], {x, \(-3\) \[Pi]/4, 3 \[Pi]/4}, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, ImageSize \[Rule] 900, Ticks \[Rule] {{\(-\[Pi]\)/2, \[Pi]/2}, {\(-40\), \(-20\), 20, 40}}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = Show[% //. Line[{a___, b : {_, y1_}, c : {_, y2_}, d___}] \[RuleDelayed] Sequence[Line[{a, b}], Line[{c, d}]] /; Abs[y1 - y2] > 20, PlotRange \[Rule] {\(-58\), 58}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[p1, p2, AxesStyle \[Rule] RGBColor[0, 0.5, 0], ImageSize \[Rule] 1100, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 0.18 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["In this exercise it is necessary to use", FontVariations->{"CompatibilityType"->0}], " the functions Re, Im and Arg as introduced in Example 3.1.1 above. In the \ illustration ", StyleBox["Mathematica", FontSlant->"Italic"], " plots vertical lines without due cause; therefore these are removed by \ hand." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(re[z_] := \(1\/2\) \((z + Conjugate[z])\)\), "\[IndentingNewLine]", \(im[z_] := \(1\/\(2 \[ImaginaryI]\)\) \((z - Conjugate[z])\)\), "\[IndentingNewLine]", \(\(\(arg[z_] := 2 ArcTan[ im[z]\/\(\@\(re[z]\^2 + im[z]\^2\) + \ re[z]\)]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(f[z_] := \(-Log[1 - z]\)\), "\[IndentingNewLine]", \(Integrate[\[Sum]\+\(k = 0\)\%\[Infinity] z\^k, z] \[Equal] f[z]\), "\[IndentingNewLine]", \(\[Sum]\+\(k = 1\)\%\[Infinity] z\^k\/k \[Equal] f[z]\), "\[IndentingNewLine]", \(FullSimplify[\[Sum]\+\(k = 1\)\%\[Infinity] Cos[k\ \[Alpha]]\/k \ \[Equal] re[f[\[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)]], {0 < \[Alpha] < 2 \[Pi]}]\), "\[IndentingNewLine]", \(FullSimplify[ re[f[\[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)]] \[Equal] \(-Log[ 2 Sin[\[Alpha]\/2]]\), 0 < \[Alpha] < 2 \[Pi]]\)}], "Input"], Cell["\<\ The sum of cosines is obviously real. Let us check the imaginary \ part anyway.\ \>", "Text"], Cell[BoxData[ \(\(\(claim = FullSimplify[ im[\[Sum]\+\(k = 1\)\%\[Infinity] Cos[k\ \[Alpha]]\/k] \[Equal] im[\(-Log[2 Sin[\[Alpha]\/2]]\)], 0 < \[Alpha] < 2 \[Pi]]\[IndentingNewLine] FullSimplify[claim /. Arg \[Rule] arg, 0 < \[Alpha] < 2 \[Pi]]\)\(\ \)\)\)], "Input"], Cell["Now we verify that the sum of sines is a sawtooth function.", "Text"], Cell[BoxData[{ \(Off[General::spell]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(sum = \[Sum]\+\(k = 1\)\%\[Infinity] Sin[k\ \[Alpha]]\/k\), "\ \[IndentingNewLine]", \(FullSimplify[ComplexExpand[sum] /. Arg \[Rule] arg, 0 < \[Alpha] < 2 \[Pi]]\), "\[IndentingNewLine]", \(\(Plot[Evaluate[sum], {\[Alpha], \(-2\) \[Pi], 4 \[Pi]}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 1200, Ticks \[Rule] {{\(-2\) \[Pi], \(-\[Pi]\), \[Pi], 2 \[Pi], 3 \[Pi], 4 \[Pi]}, {\(-\[Pi]\)/2, \[Pi]/2}}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[% //. Line[{a___, b : {_, y1_}, c : {_, y2_}, d___}] \[RuleDelayed] Sequence[Line[{a, b}], Line[{c, d}]] /; Abs[y1 - y2] > 1, PlotRange \[Rule] {\(-1.7\), 1.7}, DisplayFunction \[Rule] $DisplayFunction];\)\), \ "\[IndentingNewLine]", \(Limit[sum, \[Alpha] \[Rule] 0, Direction \[Rule] \(-1\)]\), "\[IndentingNewLine]", \(Limit[sum, \[Alpha] \[Rule] 2 \[Pi]\ , Direction \[Rule] 1]\), "\[IndentingNewLine]", \(FullSimplify[Tan[sum], 0 < \[Alpha] < 2 \[Pi]]\), "\[IndentingNewLine]", \(saw = 1/2 \((\[Pi] - \[Alpha])\)\), "\[IndentingNewLine]", \(FullSimplify[Tan[sum] \[Equal] Tan[saw], 0 < \[Alpha] < 2 \[Pi]]\)}], "Input"], Cell[TextData[{ "The conclusion sum = saw now follows from the fact that both functions sum \ and saw take values in the interval ] -", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], ",", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], " [ , on which one may invert the tangent by means of the arctangent." }], "Text"], Cell[BoxData[{ \(Simplify[ ArcTan[Tan[\[Alpha]]], \(-\(\[Pi]\/2\)\) < \[Alpha] < \[Pi]\/2]\), "\ \[IndentingNewLine]", \(Simplify[\(-\(\[Pi]\/2\)\) < saw < \[Pi]\/2, 0 < \[Alpha] < 2 \[Pi]]\), "\[IndentingNewLine]", \(Simplify[ArcTan[Tan[saw]] \[Equal] saw, 0 < \[Alpha] < 2 \[Pi]]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 1: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 1.15 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex1.15.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 1.45 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[ General::spell1]\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\)\)\>\"", "]"}], "\[IndentingNewLine]", \(p = Table[Graphics[{darkblue, PointSize[0.015], Point[{i, j}]}], {i, 0, 1, 1\/2}, {j, 0, 1, 1\/2}];\), "\[IndentingNewLine]", \({\[Gamma]\[Gamma][ 0], \[Gamma]\[Gamma][1], \[Gamma]\[Gamma][2], \[Gamma]\[Gamma][ 3], \[Gamma]\[Gamma][4], \[Gamma]\[Gamma][ 5]} = \(1\/3\) {{0, 0}, {1, 0}, {1, 2}, {2, 2}, {2, 0}, {3, 0}};\), "\[IndentingNewLine]", \(\(R[0]\)[{x1_, x2_}] := {x1, x2} - {x1, x2} . {\(-1\), 1} {\(-1\), 1}\), "\[IndentingNewLine]", \(\(R[1]\)[{x1_, x2_}] := {x1, x2}\), "\[IndentingNewLine]", \(R[2] = R[1];\), "\[IndentingNewLine]", \(\(R[3]\)[{x1_, x2_}] := {x1, x2} - {x1, x2} . {1, 1} {1, 1} + \(1\/2\) {1, 1}\), "\[IndentingNewLine]", \(\(T[0]\)[{x1_, x2_}] := {x1, x2}\), "\[IndentingNewLine]", \(\(T[1]\)[{x1_, x2_}] := {x1, x2} + \(1\/2\) {0, 1}\), "\[IndentingNewLine]", \(\(T[2]\)[{x1_, x2_}] := {x1, x2} + \(1\/2\) {1, 1}\), "\[IndentingNewLine]", \(\(T[ 3]\)[{x1_, x2_}] := {x1, x2} + \(1\/2\) {1, 0}\), "\[IndentingNewLine]", \(\(\[Beta][ i_]\)[{x1_, x2_}] := \(Composition[T[i], R[i]]\)[\(1\/2\) {x1, x2}]\), "\[IndentingNewLine]", \(gamma = Table[\[Gamma]\[Gamma][j], {j, 0, 5}];\), "\[IndentingNewLine]", RowBox[{\(\[Gamma] = ListPlot[gamma, PlotJoined \[Rule] True, Ticks \[Rule] None, Axes \[Rule] False, PlotStyle \[Rule] RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\), " "}], "\[IndentingNewLine]", \(gam[i_] := Table[\(\[Beta][i]\)[\[Gamma]\[Gamma][j]], {j, 0, 5}]\), "\[IndentingNewLine]", RowBox[{\(beta[i_] := ListPlot[gam[i], PlotJoined \[Rule] True, Ticks \[Rule] None, Axes \[Rule] False, PlotStyle \[Rule] RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{\(Show[p, \[Gamma], AspectRatio \[Rule] Automatic, Background \[Rule] lightblue];\), " "}], "\[IndentingNewLine]", RowBox[{\(Show[p, Table[beta[i], {i, 0, 3}], AspectRatio \[Rule] Automatic, Background \[Rule] lightblue];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Clear[\[Gamma], gamma]\), "\[IndentingNewLine]", \(p = Table[Graphics[{darkblue, PointSize[0.015], Point[{i, j}]}], {i, 0, 1, 1\/2}, {j, 0, 1, 1\/2}];\), "\[IndentingNewLine]", \(\[Gamma][ t_] := {0, 0} /; t < 0;\), "\[IndentingNewLine]", \(\[Gamma][ t_] := {t 5\/3, 0} /; 0 \[LessEqual] t < 1\/5;\), "\[IndentingNewLine]", \(\[Gamma][ t_] := {1\/3, \(10\/3\) \((t - 1\/5)\)} /; 1\/5 \[LessEqual] t < 2\/5;\), "\[IndentingNewLine]", \(\[Gamma][ t_] := {1\/3 + \(5\/3\) \((t - 2\/5)\), 2\/3} /; 2\/5 \[LessEqual] t < 3\/5;\), "\[IndentingNewLine]", \(\[Gamma][ t_] := {2\/3, 2\/3 - \(10\/3\) \((t - 3\/5)\)} /; 3\/5 \[LessEqual] t < 4\/5;\), "\[IndentingNewLine]", \(\[Gamma][ t_] := {2\/3 + \(5\/3\) \((t - 4\/5)\), 0} /; 4\/5 \[LessEqual] t \[LessEqual] 1;\), "\[IndentingNewLine]", \(\[Gamma][t_] := {1, 0} /; t > 1;\), "\[IndentingNewLine]", \(gamma[0] = ParametricPlot[\[Gamma][t], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(Show[ gamma[0], p, DisplayFunction \[Rule] $DisplayFunction];\), "\[IndentingNewLine]", \ \(\(\[CapitalPhi][\[Gamma]_]\)[ t_] := \(1\/2\) {\(\[Gamma][4 t]\)[\([2]\)], \(\[Gamma][ 4 t]\)[\([1]\)]} /; 0 \[LessEqual] t < 1\/4;\), "\[IndentingNewLine]", \(\(\[CapitalPhi][\[Gamma]_]\)[ t_] := \(1\/2\) {\(\[Gamma][4 t - 1]\)[\([1]\)], 1 + \(\[Gamma][4 t - 1]\)[\([2]\)]} /; 1\/4 \[LessEqual] t < 2\/4;\), "\[IndentingNewLine]", \(\(\[CapitalPhi][\[Gamma]_]\)[ t_] := \(1\/2\) {1 + \(\[Gamma][4 t - 2]\)[\([1]\)], 1 + \(\[Gamma][4 t - 2]\)[\([2]\)]} /; 2\/4 \[LessEqual] t < 3\/4;\), "\[IndentingNewLine]", \(\(\[CapitalPhi][\[Gamma]_]\)[ t_] := \(1\/2\) {2 - \(\[Gamma][4 t - 3]\)[\([2]\)], 1 - \(\[Gamma][4 t - 3]\)[\([1]\)]} /; 3\/4 \[LessEqual] t \[LessEqual] 1;\), "\[IndentingNewLine]", \(gamma[i_] := ParametricPlot[\(Nest[\[CapitalPhi], \[Gamma], i]\)[t], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 1200, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", RowBox[{\(Table[ Show[gamma[i], p, ImageSize \[Rule] \((400 + 100\ i)\), AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] $DisplayFunction], {i, 2, 4, 1}];\), " "}], "\[IndentingNewLine]", \(gamma[6] = ParametricPlot[ Evaluate[\(Nest[\[CapitalPhi], \[Gamma], 6]\)[t]], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 8000, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", RowBox[{\(Show[gamma[6], p, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\), " "}], "\[IndentingNewLine]", RowBox[{\( (*\ Computing\ these\ final\ two\ pictures\ took\ 12\ hours\ on\ my\ \ computer . \ The\ results\ are\ displayed . \ \ \ \[IndentingNewLine]gamma[ 7] = ParametricPlot[ Evaluate[\(Nest[\[CapitalPhi], \[Gamma], 7]\)[t]], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 39999, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[ gamma[7], ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction]; \ \ \[IndentingNewLine]gamma[8] = ParametricPlot[ Evaluate[\(Nest[\[CapitalPhi], \[Gamma], 8]\)[t]], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 99999, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[ gamma[8], ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\ *) \), " "}]}], "Input"], Cell["\<\ Here is a variation where all the approximating curves have no \ self-intersection.\ \>", "Text"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(p = Table[Graphics[{darkblue, PointSize[0.005], Point[{i, j}]}], {i, 0, 1, 1\/2}, {j, 0, 1, 1\/2}];\)\), "\[IndentingNewLine]", \(\(\[Gamma][t_] := {0, 0} /; t < 0;\)\), "\[IndentingNewLine]", \(\(\[Gamma][t_] := {1\/4, 1\/4 + \(6\/4\) t} /; 0 \[LessEqual] t < 1\/3;\)\), "\[IndentingNewLine]", \(\(\[Gamma][t_] := {1\/4 + \(6\/4\) \((t - 1\/3)\), 3\/4} /; 1\/3 \[LessEqual] t < 2\/3;\)\), "\[IndentingNewLine]", \(\(\[Gamma][t_] := {3\/4, 3\/4 - \(6\/4\) \((t - 2\/3)\)} /; 2\/3 \[LessEqual] t < 1;\)\), "\[IndentingNewLine]", \(\(\[Gamma][t_] := {1, 0} /; t > 1;\)\), "\[IndentingNewLine]", \(\(\(\[CapitalPhi][\[Gamma]_]\)[ t_] := \(1\/2\) {\(\[Gamma][4 t]\)[\([2]\)], \(\[Gamma][ 4 t]\)[\([1]\)]} /; 0 \[LessEqual] t < 1\/4;\)\), "\[IndentingNewLine]", \(\(\(\[CapitalPhi][\[Gamma]_]\)[ t_] := \(1\/2\) {\(\[Gamma][4 t - 1]\)[\([1]\)], 1 + \(\[Gamma][4 t - 1]\)[\([2]\)]} /; 1\/4 \[LessEqual] t < 2\/4;\)\), "\[IndentingNewLine]", \(\(\(\[CapitalPhi][\[Gamma]_]\)[ t_] := \(1\/2\) {1 + \(\[Gamma][4 t - 2]\)[\([1]\)], 1 + \(\[Gamma][4 t - 2]\)[\([2]\)]} /; 2\/4 \[LessEqual] t < 3\/4;\)\), "\[IndentingNewLine]", \(\(\(\(\[CapitalPhi][\[Gamma]_]\)[ t_] := \(1\/2\) {2 - \(\[Gamma][4 t - 3]\)[\([2]\)], 1 - \(\[Gamma][4 t - 3]\)[\([1]\)]} /; 3\/4 \[LessEqual] t \[LessEqual] 1;\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(gamma[n_] := ParametricPlot[ Evaluate[\(Nest[\[CapitalPhi], \[Gamma], n]\)[t]], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 11999, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Table[ Show[gamma[n], p, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction], {n, 0, 4}];\)\), "\[IndentingNewLine]", \(\(gamma[5] := ParametricPlot[ Evaluate[\(Nest[\[CapitalPhi], \[Gamma], 5]\)[t]], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 23999, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[gamma[5], p, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\)\ \), \ "\[IndentingNewLine]", \(\(gamma[6] = ParametricPlot[ Evaluate[\(Nest[\[CapitalPhi], \[Gamma], 6]\)[t]], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 59999, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[gamma[6], p, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\)\[IndentingNewLine]\), "\ \[IndentingNewLine]", \( (*\ Computing\ the\ final\ two\ pictures\ took\ a\ substantial\ amount\ of\ \ time\ on\ my\ computer . \ The\ results\ are\ displayed . \ \[IndentingNewLine]gamma[7] = ParametricPlot[ Evaluate[\(Nest[\[CapitalPhi], \[Gamma], 7]\)[t]], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 99999, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[ gamma[7], ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction]; \ \[IndentingNewLine]gamma[ 8] = ParametricPlot[ Evaluate[\(Nest[\[CapitalPhi], \[Gamma], 8]\)[t]], {t, 0, 1}, Ticks \[Rule] None, Axes \[Rule] False, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 119999, PlotStyle \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[ gamma[8], ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\ *) \)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 1.46 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(p[1]\)[t_] := 0\), "\[IndentingNewLine]", \(\(p[k_]\)[ t_] := \(1\/2\) \((t\^2 + 2 \( p[k - 1]\)[ t] - \(p[k - 1]\)[t]\^2)\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Do[Print[Simplify[Expand[\(p[k]\)[t]]]], {k, 10}]\), "\[IndentingNewLine]", \(Table[\(p[k]\)[0], {k, 10}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Table[ Simplify[Expand[ 2 \((Abs[t] - \(p[k + 1]\)[t])\) - Abs[t] \((2 - Abs[t])\) + \(p[k]\)[t] \((2 - \(p[k]\)[t])\)], Im[t] \[Equal] 0], {k, 10}]\), "\[IndentingNewLine]", \(Table[ Simplify[Expand[ 2 \((\(p[k + 1]\)[t] - \(p[k]\)[t])\) - t\^2 + \(p[k]\)[t]\^2]], {k, 10}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(q[k_] := Plot[\(p[k]\)[t], {t, \(-1\), 1}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, PlotPoints \[Rule] 1000, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(a = Plot[Abs[t], {t, \(-1\), 1}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] darkblue, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, PlotPoints \[Rule] 1000, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(t[k_] := Show[a, q[k], DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 500];\)\), "\[IndentingNewLine]", \(\(Table[t[k], {k, 2, 10}];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 2: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 2.7 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex2.7.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.10 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(rmin = 0.005;\)\), "\n", \(\(rmax = 1;\)\), "\n", \(\(\[Alpha]min = 0;\)\), "\n", \(\(\[Alpha]max = 2 \[Pi];\)\), "\[IndentingNewLine]", \(\(nr = 89;\)\), "\n", \(\(n\[Alpha] = 155;\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\n", \(f[x1_, x2_] := \(x1\ x2\^3\)\/\(x1\^2 + x2\^4\)\), "\[IndentingNewLine]", \(x1[r_, \[Alpha]_] := r\ Cos[\[Alpha]]\), "\n", \(x2[r_, \[Alpha]_] := r\ Sin[\[Alpha]]\), "\n", \(x3[r_, \[Alpha]_] := f[x1[r, \[Alpha]], x2[r, \[Alpha]]]\), "\n", \(\(ParametricPlot3D[{x1[r, \[Alpha]], x2[r, \[Alpha]], x3[r, \[Alpha]]} // Evaluate, {r, rmin, rmax}, {\[Alpha], \[Alpha]min, \[Alpha]max}, PlotPoints \[Rule] {nr, n\[Alpha]}, ViewPoint \[Rule] {1, 1, 1}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, PlotRange \[Rule] {{\(-1\), 1}, {\(-1\), 1}, {\(-0.45\), 0.45}}, Background \[Rule] lightblue, ImageSize \[Rule] 1200];\)\), "\[IndentingNewLine]", \(Limit[Abs[f[x2\^2, x2]]\/\@\(x2\^4 + x2\^2\), x2 \[Rule] 0]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.11 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(rmin = 0.001;\)\), "\n", \(\(rmax = 1;\)\), "\n", \(\(\[Alpha]min = 0;\)\), "\n", \(\(\[Alpha]max = 2 \[Pi];\)\), "\[IndentingNewLine]", \(\(nr = 89;\)\), "\n", \(\(n\[Alpha] = 155;\)\), "\[IndentingNewLine]", \(f[x1_, x2_] := x1\^3\/\(x1\^2 + x2\^2\)\), "\[IndentingNewLine]", \(D1f[x1_, x2_] := \[PartialD]\_x1\ f[x1, x2]\), "\[IndentingNewLine]", \(D2f[x1_, x2_] := \[PartialD]\_x2\ f[x1, x2]\), "\[IndentingNewLine]", \(x1[r_, \[Alpha]_] := r\ Cos[\[Alpha]]\), "\n", \(x2[r_, \[Alpha]_] := r\ Sin[\[Alpha]]\), "\n", \(x3[r_, \[Alpha]_] := f[x1[r, \[Alpha]], x2[r, \[Alpha]]]\), "\n", \(\(ParametricPlot3D[{x1[r, \[Alpha]], x2[r, \[Alpha]], x3[r, \[Alpha]]} // Evaluate, {r, rmin, rmax}, {\[Alpha], \[Alpha]min, \[Alpha]max}, PlotPoints \[Rule] {nr, n\[Alpha]}, ViewPoint \[Rule] {0.8, 1, 3}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, PlotRange \[Rule] {{\(-0.5\), 0.5}, {\(-0.5\), 0.5}, {\(-0.25\), 0.25}}, Background \[Rule] lightblue, ImageSize \[Rule] 1200];\)\), "\[IndentingNewLine]", \(D1f[x\_1, x\_2]\), "\[IndentingNewLine]", \(Simplify[ D1f[x1, x2] \[Equal] 1 + \(x2\^2\) \(x1\^2 - x2\^2\)\/\((x1\^2 + x2\^2)\)\^2]\), "\ \[IndentingNewLine]", \(D2f[x\_1, x\_2]\), "\[IndentingNewLine]", \(Clear[x3]\), "\n", \(x3[r_, \[Alpha]_] := D1f[y1, y2] /. {y1 -> x1[r, \[Alpha]], y2 -> x2[r, \[Alpha]]}\), "\[IndentingNewLine]", \(\(ParametricPlot3D[{x1[r, \[Alpha]], x2[r, \[Alpha]], x3[r, \[Alpha]]} // Evaluate, {r, rmin, rmax}, {\[Alpha], \[Alpha]min, \[Alpha]max}, PlotPoints \[Rule] {nr, n\[Alpha]}, ViewPoint \[Rule] {0.8, 1, 3}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, PlotRange \[Rule] {{\(-0.5\), 0.5}, {\(-0.5\), 0.5}, {0, 1.3}}, Background \[Rule] lightblue, ImageSize \[Rule] 1200];\)\), "\[IndentingNewLine]", \(Clear[x3]\), "\n", \(x3[r_, \[Alpha]_] := D2f[y1, y2] /. {y1 -> x1[r, \[Alpha]], y2 -> x2[r, \[Alpha]]}\), "\[IndentingNewLine]", \(\(ParametricPlot3D[{x1[r, \[Alpha]], x2[r, \[Alpha]], x3[r, \[Alpha]]} // Evaluate, {r, rmin, rmax}, {\[Alpha], \[Alpha]min, \[Alpha]max}, PlotPoints \[Rule] {nr, n\[Alpha]}, ViewPoint \[Rule] {0.8, 1, 3}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, PlotRange \[Rule] {{\(-0.5\), 0.5}, {\(-0.5\), 0.5}, {\(-3\)/8 Sqrt[3] - 0.1, 3/8 Sqrt[3] + 0.1}}, Background \[Rule] lightblue, ImageSize \[Rule] 1200];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.17", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(f[x1_, x2_] := x1\ g[x2\/x1] + h[x2\/x1]\), "\[IndentingNewLine]", \(FullSimplify[ x1^2\ \[PartialD]\_\(x1, x1\)f[x1, x2] + 2 x1\ x2\ \[PartialD]\_\(x1, x2\)f[x1, x2] + x2^2\ \[PartialD]\_\(x2, x2\)f[x1, x2]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercises 2.19 and 2.20 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the computations in Section 2.4 Chain rule above. ", "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.22 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex2.22.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.25 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex2.25.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercises 2.29", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " produces many spurious solutions, due to limitations in machine \ precision. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Divide::infy]\), "\[IndentingNewLine]", \(Off[\[Infinity]::indet]\), "\[IndentingNewLine]", \(Off[FindRoot::cvmit]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(f[x_] := Log[1 + \[ExponentialE]\^x]\), "\[IndentingNewLine]", \(Simplify[\(f'\)[x] < 1, Im[x] \[Equal] 0]\), "\[IndentingNewLine]", \(Simplify[f[x] \[Equal] x + f[\(-x\)], Im[x] \[Equal] 0]\), "\[IndentingNewLine]", \(\(g = Plot[f[x], {x, \(-1\), 10}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, PlotPoints \[Rule] 1000, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(b = Plot[x, {x, \(-1\), 10}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] darkblue, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, PlotPoints \[Rule] 1000, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[b, g, DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 500];\)\), "\[IndentingNewLine]", \(r[0] = FindRoot[f[\(-x\)] \[Equal] 0, {x, 10}, WorkingPrecision \[Rule] $MachinePrecision]\), "\[IndentingNewLine]", \ \(f[x] - x /. r[0]\), "\[IndentingNewLine]", \(r[n_] := FindRoot[f[\(-x\)] \[Equal] 0, {x, n}, WorkingPrecision \[Rule] $MachinePrecision]\), "\[IndentingNewLine]", \ \(Table[f[x] - x /. r[n], {n, 111, 130}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercises 2.30", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can compute the gradient of a function in the case of ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "2"], TraditionalForm]]], "or ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "3"], TraditionalForm]]], "using a package. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(<< Calculus`VectorAnalysis`\), "\[IndentingNewLine]", \(Grad[f[x1, x2, x3]\ , Cartesian[x1, x2, x3]] /. Thread[{x1, x2, x3} \[Rule] {x\_1, x\_2, x\_3}]\), "\[IndentingNewLine]", \(f[x1_, x2_, x3_] := Log[\@\(x1\^2 + x2\^2\)]\), "\[IndentingNewLine]", \(Grad[f[x1, x2, x3]\ , Cartesian[x1, x2, x3]] /. Thread[{x1, x2} \[Rule] {x\_1, x\_2}]\), "\[IndentingNewLine]", \(f[x1_, x2_, x3_] := \(-1\)\/\@\(x1\^2 + x2\^2 + x3\^2\)\), "\[IndentingNewLine]", \ \(Grad[f[x1, x2, x3]\ , Cartesian[x1, x2, x3]] /. Thread[{x1, x2, x3} \[Rule] {x\_1, x\_2, x\_3}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.31 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex2.31.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.36 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(f[x1_, x2_] := \((x1\^2 - x2)\) \((3 x1\^2 - x2)\)\), "\[IndentingNewLine]", \(x1[r_, \[Alpha]_] := r\ Cos[\[Alpha]]\), "\n", \(x2[r_, \[Alpha]_] := r\ Sin[\[Alpha]]\), "\n", \(x3[r_, \[Alpha]_] := f[x1[r, \[Alpha]], x2[r, \[Alpha]]]\), "\[IndentingNewLine]", \(p[r_, \[Alpha]_] := {x1[r, \[Alpha]], x2[r, \[Alpha]], x3[r, \[Alpha]]}\), "\[IndentingNewLine]", \(f[t, t\^2]\), "\[IndentingNewLine]", \(f[t, 2 t\^2]\), "\[IndentingNewLine]", \(f[t, 3 t\^2]\), "\[IndentingNewLine]", \(f[t, 4 t\^2]\), "\[IndentingNewLine]", \(\(Plot[{t\^2, 2 t\^2, 3 t\^2, 4 t\^2}, {t, \(-0.3\), 0.3}, PlotRange \[Rule] {\(-0.05\), 0.3}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[p[r, \[Alpha]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {r, 0, 2}, {\[Alpha], 0, 2 \[Pi]}, PlotPoints \[Rule] 151, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, ViewPoint \[Rule] 2 {0.7, 1.1, 0.4}, PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(-0.9\), 0.9}, {\(-0.5\), 0.6}}, Boxed \[Rule] False, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 1100];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(Plot[f[0, t], {t, \(-0.5\), 0.5}, PlotRange \[Rule] {\(-0.05\), 0.3}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(\(Plot[f[t, 2 t\^2], {t, \(-0.5\), 0.5}, PlotRange \[Rule] {\(-0.1\), 0.05}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] True, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(g[x1_] := f[x1, \[Lambda]\ x1]\), "\[IndentingNewLine]", \(\(\[Lambda] = 2;\)\), "\[IndentingNewLine]", \(\(Plot[g[x\_1], {x\_1, \(-1\), 3}, PlotRange \[Rule] {\(-3\), 1}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(Clear[\[Lambda]]\), "\[IndentingNewLine]", \(sol = Flatten[Solve[\[PartialD]\_\(x\_1\)\ g[x\_1] \[Equal] 0, x\_1]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(x1 = x\_1 /. sol[\([1]\)]\), "\[IndentingNewLine]", \(g[x1]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(x3 = Simplify[x\_1 /. sol[\([3]\)]]\), "\[IndentingNewLine]", \(Simplify[g[x3]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(x2 = Simplify[x\_1 /. sol[\([2]\)]]\), "\[IndentingNewLine]", \(Simplify[g[x2]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.39 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex2.39.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.47 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\n", \(f[x_, y_] := x\^2 - y\), "\[IndentingNewLine]", \(\(x0 = 5.5;\)\), "\n", \(\(x00 = x0 + 1;\)\), "\[IndentingNewLine]", \(\(y0 = 2;\)\), "\n", \(\(x1 = \((x0 + y0/x0)\)/2;\)\), "\n", \(\(x11 = x1 + 1;\)\), "\n", \(\(x2 = \((x1 + y0/x1)\)/2;\)\), "\n", \(\(z0 = 5.5;\)\), "\n", \(\(z00 = z0 + 1;\)\), "\n", \(\(A = 1/\((2\ z0)\);\)\), "\n", \(\(z1 = z0 - A\ f[z0, y0];\)\), "\n", \(\(z2 = z1 - A\ f[z1, y0];\)\), "\n", \(\(z3 = z2 - A\ f[z2, y0];\)\), "\[IndentingNewLine]", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(Plot[f[x, y0], {x, Sqrt[y0] - 1, Sqrt[y0] + 5.5}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {AbsoluteThickness[0.75], RGBColor[1, 0, 0]}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{AbsoluteThickness[1], Line[{{Sqrt[y0] - 1, 0}, {Sqrt[y0] + 5, 0}}], {RGBColor[0, 0, 1], Line[{{x1, 0}, {x00, f[x0, y0] + 2\ x0\ \((x00 - x0)\)}}]}, {RGBColor[0, 0, 1], Line[{{x2, 0}, {x11, f[x1, y0] + 2\ x1\ \((x11 - x1)\)}}]}, Text["\", {x0, \(-0.5\)}], Text["\", {x1, \(- .5\)}], Text["\", {x2, \(-0.5\)}], {Dashing[{0.01, 0.01}], AbsoluteThickness[0.5], Line[{{x0, 0}, {x0, f[x0, y0]}}], Line[{{x1, 0}, {x1, f[x1, y0]}}]}}];\)\), "\n", \(\(Show[%%, %, Axes \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Plot[f[x, y0], {x, Sqrt[y0] - 1, Sqrt[y0] + 5.5}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {AbsoluteThickness[0.75], RGBColor[1, 0, 0]}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{AbsoluteThickness[1], Line[{{Sqrt[y0] - 1, 0}, {Sqrt[y0] + 5, 0}}], {RGBColor[0, 0, 1], Line[{{z1, 0}, {z00, f[z0, y0] + 1/A\ \((z00 - z0)\)}}]}, {RGBColor[0, 0, 1], Line[{{z2, 0}, {z1, f[z1, y0]}}], Line[{{z3, 0}, {z2, f[z2, y0]}}]}, Text["\", {z0, \(- .5\)}], Text["\", {z1, \(- .5\)}], Text["\", {z2, \(- .5\)}], Text["\", {z3, \(-0.5\)}], {Dashing[{0.01, 0.01}], AbsoluteThickness[0.5], Line[{{z0, 0}, {z0, f[z0, y0]}}], Line[{{z1, 0}, {z1, f[z1, y0]}}], Line[{{z2, 0}, {z2, f[z2, y0]}}], Line[{{z3, 0}, {z3, f[z3, y0]}}]}}];\)\), "\n", \(\(Show[%%, %, Axes \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[GraphicsArray[{%, %%%%}], ImageSize \[Rule] 1100];\)\)}], "Input"], Cell[TextData[StyleBox["We give a concrete application of Newton's iteration \ method.", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]]], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(Plot[{2 x, Tan[x]}, {x, 0, \[Pi]\/2}, PlotRange \[Rule] {0, 3}, Background \[Rule] lightblue, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], AspectRatio \[Rule] Automatic, ImageSize \[Rule] 400];\)\), "\[IndentingNewLine]", \(\(Plot[{2 x, Tan[x]}, {x, 1.165561, 1.1655615}, Background \[Rule] lightblue, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], AspectRatio \[Rule] Automatic, ImageSize \[Rule] 400];\)\), "\[IndentingNewLine]", \(f[x_] := 2 x - Tan[x]\), "\[IndentingNewLine]", \(F[x_] := N[x - \(1\/\(f'\)[x]\) f[x], 150]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FixedPoint[F, 6\/5]\), "\[IndentingNewLine]", \(f[%]\), "\[IndentingNewLine]", \(FixedPointList[F, 6\/5]\), "\[IndentingNewLine]", \(Table[ Abs[%[\([k + 1]\)] - %[\([k]\)]], {k, Length[%] - 1}]\), "\[IndentingNewLine]", \(FindRoot[f[x] \[Equal] 0, {x, 6\/5}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FixedPoint[F, 1\/5]\), "\[IndentingNewLine]", \(f[%]\), "\[IndentingNewLine]", \(FixedPointList[F, 1\/5]\), "\[IndentingNewLine]", \(Table[Abs[%[\([k + 1]\)] - %[\([k]\)]], {k, Length[%] - 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.79 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[\(\(\ \)\(1\)\)\/\(1 + p\ \ Cos[\[Alpha]]\), {\[Alpha], 0, \[Pi]}, Assumptions \[Rule] {\(-1\) < p < 1}]\), "\[IndentingNewLine]", \(Integrate[\ Log[1 + x\ \ Cos[\[Alpha]]]\/Cos[\[Alpha]], {\[Alpha], 0, \[Pi]}, Assumptions \[Rule] {\(-1\) < x < 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.80 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is able to evaluate only the first two of the three integrals in the \ exercise, in case of the third it is not able to produce the compact form of \ the answer. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Integrate[1\/\(1 - a\^2\ \(y\^2\) Sin[x]\^2\), {x, 0, \[Pi]\/2}, Assumptions -> {0 < a < 1, 0 < y < 1}]\), "\[IndentingNewLine]", \(Integrate[1\/\(1 - a\^2\ \(y\^2\) Sin[x]\^2\), {y, 0, 1}, Assumptions \[Rule] {0 < a < 1, 0 < x < \[Pi]\/2}]\), "\[IndentingNewLine]", \(FullSimplify[ TrigExpand[%] \[Equal] \(1\/\(2 a\ Sin[x]\)\) Log[\(1 + a\ Sin[x]\)\/\(1 - a\ Sin[x]\)], Assumptions \[Rule] {0 < a < 1, Im[x] \[Equal] 0}]\), "\[IndentingNewLine]", \(f[a_] := Integrate[ 1\/Sin[x]\ Log[\(1 + a\ Sin[x]\)\/\(1 - a\ Sin[x]\)], {x, 0, \[Pi]\/2}, Assumptions \[Rule] {0 < a < 1}]\ \), "\[IndentingNewLine]", \(f[a]\), "\[IndentingNewLine]", \(g[a_] := \[Pi]\ ArcSin[a] - f[a]\), "\[IndentingNewLine]", \(FullSimplify[g[a], Assumptions \[Rule] {0 < a < 1}]\), "\[IndentingNewLine]", \(FullSimplify[\[PartialD]\_a\ g[a], {0 < a < 1}]\), "\[IndentingNewLine]", \(g[1\/2]\), "\[IndentingNewLine]", \(\(Plot[Evaluate[g[a]], {a, 0, 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 800];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.82 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[\ \(1 - Cos[x\ t]\)\/t\^2, {t, 0, \[Infinity]}, Assumptions \[Rule] {Im[x] == 0}]\), "\[IndentingNewLine]", \(Integrate[\ Sin[x\ t]\^2\/t\^2, {t, 0, \[Infinity]}, Assumptions \[Rule] {Im[x] == 0}]\), "\[IndentingNewLine]", \(Integrate[\(\(\ \)\(Sin[x\ t]\^4\)\)\/t\^2, {t, 0, \[Infinity]}, Assumptions \[Rule] {Im[x] == 0}]\), "\[IndentingNewLine]", \(Integrate[\ Sin[x\ t]\^4\/t\^4, {t, 0, \[Infinity]}, Assumptions \[Rule] {Im[x] == 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.85 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(TrigExpand[ Integrate[Sin[x\ t]\/\(t \((1 + t\^2)\)\), {t, 0, \[Infinity]}, Assumptions \[Rule] {x > 0}]]\), "\[IndentingNewLine]", \(Integrate[Cos[x\ t]\/\(1 + t\^2\), {t, 0, \[Infinity]}, Assumptions \[Rule] {x > 0}]\), "\[IndentingNewLine]", \(Integrate[\(t\ Sin[x\ t]\)\/\(1 + t\^2\), {t, 0, \[Infinity]}, Assumptions \[Rule] {x > 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.86 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[\ Log[1 + \(x\^2\) t\^2]\/\(1 + t\^2\), {t, 0, \[Infinity]}, Assumptions \[Rule] {x \[GreaterEqual] 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.87 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \("\"\), "\[IndentingNewLine]", \(Integrate[\[ExponentialE]\^\(\(-t\^2\) - x\^2/t\^2\), {t, 0, \[Infinity]}, Assumptions \[Rule] {x > 0}]\), "\[IndentingNewLine]", \("\"\), "\[IndentingNewLine]", \(\(\@x\) Integrate[\[ExponentialE]\^\(\(-x\) \((t\^2 + t\^\(-2\))\)\), {t, 0, \[Infinity]}, Assumptions \[Rule] {x > 0}]\), "\[IndentingNewLine]", \("\"\), "\[IndentingNewLine]", \(Integrate[\(1\/\@t\) \[ExponentialE]\^\(\(-a\^2\) t - b\^2/t\), {t, 0, \[Infinity]}, Assumptions \[Rule] {a > 0, b > 0}]\), "\[IndentingNewLine]", \(Integrate[\(1\/\(t\ \@t\)\) \[ExponentialE]\^\(\(-a\^2\) t - b\^2/t\), \ {t, 0, \[Infinity]}, Assumptions \[Rule] {a > 0, b > 0}]\), "\[IndentingNewLine]", \(Integrate[ 1\/\@\[Pi]\ \[ExponentialE]\^\(-t\)\/\@t\ \[ExponentialE]\^\(\(-x\^2\)/\ \((4 t)\)\), {t, 0, \[Infinity]}, Assumptions \[Rule] {x > 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.88 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[N::meprec]\), "\[IndentingNewLine]", \(Simplify[Integrate[ArcTan[x\ t]\/\(t\ \@\(1 - t\^2\)\), {t, 0, 1}], x > 0]\), "\[IndentingNewLine]", \(FullSimplify[ TrigExpand[ArcSinh[x]] \[Equal] Log[x + \@\(1 + x\^2\)]]\), "\[IndentingNewLine]", \(FullSimplify[ TrigExpand[\[Pi]\/2\ ArcSinh[1]] \[Equal] \[Pi]\/2\ Log[ 1 + \@2]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.89 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[N::meprec]\), "\[IndentingNewLine]", \(Integrate[1\/\(1 + \(t\^2\) y\^2\), {y, 0, 1}, Assumptions \[Rule] {Im[t] == 0, t \[NotEqual] 0}]\), "\[IndentingNewLine]", \(Limit[%, t \[Rule] 0]\), "\[IndentingNewLine]", \(\[Pi]\/2\ Integrate[ 1\/\@\(1 + y\^2\), {y, 0, 1}]\), "\[IndentingNewLine]", \(FullSimplify[\ TrigExpand[%] \[Equal] \(\[Pi]\/2\) Log[1 + \@2]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2.90 (Airy function) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(AiryAi''\)[x] - x\ AiryAi[x]\), "\[IndentingNewLine]", \(AiryAi[0]\), "\[IndentingNewLine]", \(FullSimplify[ AiryAi[0] - Gamma[1\/3]\/\(2 \[Pi]\ 3\^\(1\/6\)\)]\), "\[IndentingNewLine]", \(\(AiryAi'\)[0]\), "\[IndentingNewLine]", \(FullSimplify[\(AiryAi'\)[ 0] + \(\(3\^\(1\/6\)\) Gamma[2\/3]\)\/\(\(2\) \(\[Pi]\)\(\ \ \)\)]\), "\[IndentingNewLine]", \(FullSimplify[ Series[AiryAi[x], {x, 0, 11}] - AiryAi[0] \((1 + \(1\/\(3!\)\) x\^3 + \(\(1\ 4\)\/\(6!\)\) x\^6 + \(\(1\ 4\ 7\)\/\(9!\)\) x\^9)\) - \(AiryAi'\)[ 0] \((x + \(2\/\(4!\)\) x\^4 + \(\(2\ 5\)\/\(7!\)\) x\^7 + \(\(2\ 5\ 8\)\/\(10!\)\) x\^10)\)]\), "\[IndentingNewLine]", \(\(Plot[AiryAi[x], {x, \(-30\), 7}, PlotPoints \[Rule] 10000, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 700];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 3: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 3.1 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Flatten[ Solve[{y\_1 + y\_2 == x\_1, y\_1\/y\_2 == x\_2}, {y\_1, y\_2}]] /. {Rule \[Rule] Equal}\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.2 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Flatten[ Solve[{y\_1\ y\_2 == x\_1, \((1 - y\_1)\) y\_2 == x\_2}, {y\_1, y\_2}]] /. {Rule \[Rule] Equal}\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.3 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Flatten[ Simplify[ Solve[{x\_1 + x\_2 \[Equal] y\_1, x\_1 - x\_2 \[Equal] y\_2}, {x\_1, x\_2}]]] /. {Rule \[Rule] Equal}\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.7 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(t0 = \ \[Pi]/4;\)\), "\n", \(\(s0 = 0.3\ \[Pi];\)\), "\n", \(\(r = 1;\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\n", \(\(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(p1 = Graphics3D[{{RGBColor[1, 0, 0], AbsolutePointSize[5], Point[{0, 0, r\ Sin[t0]}], Point[{r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], r\ Sin[t0]}], Point[{0, r\ Cos[t0]\ Sin[s0], 0}], Point[{r\ Cos[t0]\ Cos[s0], 0, 0}]}, \[IndentingNewLine]{darkblue, AbsoluteThickness[1], Line[{{0, 0, r\ Sin[t0]}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], r\ Sin[t0]}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], 0}}], Line[{{r\ Cos[t0]\ Cos[s0], 0, 0}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], 0}, {0, r\ Cos[t0]\ Sin[s0], 0}}], Line[{{0, 0, 0}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], 0}}], Line[{{0, 0, 0}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], r\ Sin[t0]}}]}, Text["\", {0, 0, r\ Sin[t0]}], Text["\", {3\ r/4\ Cos[t0]\ Cos[s0], 3\ r/4\ Cos[t0]\ Sin[s0], 3\ r/4\ Sin[t0]}], Text["\", {0, r\ Cos[t0]\ Sin[s0], 0}], Text["\", {r\ Cos[t0]\ Cos[s0], 0, 0}], Text["\", {0.25\ Cos[s0/2], 0.25\ Sin[s0/2], 0}], Text["\", {0.25\ Cos[t0/2]\ Cos[s0], 0.25\ Cos[t0/2]\ Sin[s0], 0.25\ Sin[t0/2]}]}];\)\), "\[IndentingNewLine]", \(\(p2 = Graphics3D[{{RGBColor[0, 0.5, 0], AbsoluteThickness[0.75], Line[{{1, 0, 0}, {0, 0, 0}, {0, 1, 0}}], Line[{{0, 0, 0}, {0, 0, 1}}]}, {darkblue, AbsoluteThickness[1], Line[{{0, 0, r\ Sin[t0]}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], r\ Sin[t0]}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], 0}}], Line[{{r\ Cos[t0]\ Cos[s0], 0, 0}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], 0}, {0, r\ Cos[t0]\ Sin[s0], 0}}], Line[{{0, 0, 0}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], 0}}], Line[{{0, 0, 0}, {r\ Cos[t0]\ Cos[s0], r\ Cos[t0]\ Sin[s0], r\ Sin[t0]}}]}, Text["\", {0, 0, r\ Sin[t0]}], Text["\", {3\ r/4\ Cos[t0]\ Cos[s0], 3\ r/4\ Cos[t0]\ Sin[s0], 3\ r/4\ Sin[t0]}], Text["\", {0, r\ Cos[t0]\ Sin[s0], 0}], Text["\", {r\ Cos[t0]\ Cos[s0], 0, 0}], Text["\", {0.25\ Cos[s0/2], 0.25\ Sin[s0/2], 0}], Text["\", {0.25\ Cos[t0/2]\ Cos[s0], 0.25\ Cos[t0/2]\ Sin[s0], 0.25\ Sin[t0/2]}]}];\)\), "\[IndentingNewLine]", \(\(p3 = ParametricPlot3D[ Append[{0.25\ Cos[s], 0.25\ Sin[s], 0}, RGBColor[1, 0, 0]], {s, 0, s0}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot3D[ Append[{0.25\ Cos[t]\ Cos[s0], 0.25\ Cos[t]\ Sin[s0], 0.25\ Sin[t]}, RGBColor[1, 0, 0]], {t, 0, t0}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[p1, p2, p3, p4, Boxed \[Rule] False, ViewPoint \[Rule] {200000, 100000, 100000}, PlotRange \[Rule] {{0, 0.7}, {0, 1}, {0, 1}}, DefaultFont \[Rule] 12. , Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.11 (Confocal coordinates) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex3.11.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Exercises 3.12 and 3.14 (Gradient and divergence in arbitrary \ coordinates) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can compute the gradient of a function and the divergence of a vector \ field in certain special coordinates in the case of ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "3"], TraditionalForm]]], " using a package. Below we give an example." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(<< Calculus`VectorAnalysis`\), "\[IndentingNewLine]", \(\(SetCoordinates[ProlateSpheroidal];\)\), "\[IndentingNewLine]", \({CoordinateSystem, Coordinates[]}\), "\[IndentingNewLine]", \(CoordinatesFromCartesian[{x, y, z}, ProlateSpheroidal]\ \ \ \), "\[IndentingNewLine]", \(Grad[f[x, e, p], ProlateSpheroidal[x, e, p]]\), "\[IndentingNewLine]", \(Laplacian[f[x, e, p], ProlateSpheroidal[x, e, p]]\), "\[IndentingNewLine]", \(Div[{f\_1[x, e, p], f\_2[x, e, p], f\_3[x, e, p]}, ProlateSpheroidal[x, e, p]]\), "\[IndentingNewLine]", \(Curl[{f\_1[x, e, p], f\_2[x, e, p], f\_3[x, e, p]}, ProlateSpheroidal[x, e, p]]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.20 (Diffeomorphism of triangle onto square) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(y1[x1_, x2_] := ArcTan[x1\ \@\(\(1 - x2\^2\)\/\(1 - x1\^2\)\)]\), \ "\[IndentingNewLine]", \(y2[x1_, x2_] := y1[x2, x1]\), "\[IndentingNewLine]", \(y2[x\_1, x\_2]\), "\[IndentingNewLine]", \(PowerExpand[ Simplify[{Sin[y1[x\_1, x\_2]]\/Cos[y2[x\_1, x\_2]], Sin[y2[x\_1, x\_2]]\/Cos[y1[x\_1, x\_2]]}, {Im[x\_1] \[Equal] 0, Im[x\_2] \[Equal] 0}]]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.22 (Wave equation in one spatial variable) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(\(u[x_, t_] = \(1\/2\) \((f[x + t] + f[x - t])\) + \(1\/2\) \(\[Integral]\_\(x - t\)\%\(x + t\)g[ y] \[DifferentialD]y\);\)\), "\[IndentingNewLine]", \({u[x, 0], \[PartialD]\_t\ u[x, t]} /. t \[Rule] 0\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_\(t, t\)u[x, t] - \[PartialD]\_\(x, x\)u[x, t]]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.31 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(f[x_, y_] := Sin[x\^2 + y] - 2 x\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[PartialD]\_x\ f[x, y] /. {x \[Rule] 0, y \[Rule] 0}\), "\[IndentingNewLine]", \(\(-\(1\/\[PartialD]\_x\ f[x, y]\)\) \[PartialD]\_y\ f[x, y] /. {x \[Rule] 0, y -> 0}\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(fy[y_] := f[x[y], y]\), "\[IndentingNewLine]", \(Collect[\(fy'\)[y] /. {x[y] \[Rule] x, \(x'\)[y] \[Rule] x'}, x']\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Expand[ 1/2 \( fy''\)[y] /. {Sin[x[y]^2 + y] \[Rule] 2 x, x[y] \[Rule] x, \(x'\)[y] \[Rule] x', \(x''\)[y] \[Rule] x''}]\), "\[IndentingNewLine]", \(Flatten[ Solve[\(fy''\)[y] \[Equal] 0 /. {tf[y] \[Rule] 0, x[y] \[Rule] 0, \(x'\)[y] \[Rule] 1/2, y \[Rule] 0, \(x''\)[y] \[Rule] x''}, x'']] /. {Rule \[Rule] Equal}\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.36 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See also the file Ex3.36.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(F[x_, a_] := x - Integrate[f[t], {t, x, a}]\), "\[IndentingNewLine]", \(\[PartialD]\_x\ F[x, a] /. x \[Rule] 0\), "\[IndentingNewLine]", \(\(-\((\[PartialD]\_x\ F[x, a])\)\^\(-1\)\) \[PartialD]\_a\ F[x, a] /. {x \[Rule] 0, a \[Rule] b}\), "\[IndentingNewLine]", \(f[t_] := 2 t - 1\), "\[IndentingNewLine]", \(Flatten[ Simplify[Solve[F[x, a] \[Equal] 0, x]]] /. {Rule \[Rule] Equal}\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.43 (Cardioid) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(g[x_, y_] := x\^4 - 4 x\^3 + 2 \( y\^2\) x\^2 - 4 \( y\^2\) x + y\^4 - 4 y\^2\), "\[IndentingNewLine]", \(Eliminate[{\@\(x\^2 + y\^2\) == 2 \((1 + Cos[\[Alpha]])\), x\/\@\(x\^2 + y\^2\) == Cos[\[Alpha]]}, Cos[\[Alpha]]]\), "\[IndentingNewLine]", \(\(\(Solve[g[x, y] \[Equal] 0, x]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(<< Graphics`\), "\[IndentingNewLine]", \(r[\[Alpha]_] := 2 \((1 + Cos[\[Alpha]])\)\), "\[IndentingNewLine]", \(\(\(p1 = PolarPlot[r[\[Alpha]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Axes \[Rule] True, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(<< Graphics`ImplicitPlot`\), "\[IndentingNewLine]", \(\(\(p2 = ImplicitPlot[g[x, y] \[Equal] 0, {x, \(-1\), 4}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Axes \[Rule] True, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(p3 = ParametricPlot[ r[\[Alpha]] {\ Cos[\[Alpha]], \ Sin[\[Alpha]]}, {\[Alpha], 0, 2\ \[Pi]}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotPoints \[Rule] 50, AspectRatio \[Rule] Automatic, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(Show[GraphicsArray[{p1, p2, p3}], ImageSize \[Rule] 1200, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Exercise 3.45 (Discriminant locus)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", RowBox[{\(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(f[x_] := y\_2\ x\^2 + y\_1\ x + y\_0\), "\n", \(Eliminate[{f[x] \[Equal] 0, \[PartialD]\_x\ f[x] \[Equal] 0}, x]\), "\n", RowBox[{\(Simplify[% /. Thread[{y\_0, y\_1, y\_2} \[Rule] {\((z\_2 + z\_0)\)/2, z\_1, \((z\_2 - z\_0)\)/2}]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"", "]"}], "\[IndentingNewLine]", \(<< Graphics`ContourPlot3D`\), "\[IndentingNewLine]", RowBox[{\(ContourPlot3D[ y1\^2 - 4 y0\ y2, {y0, \(-4\), 4}, {y1, \(-6\), 6}, {y2, \(-4\), 4}, ContourStyle \[Rule] RGBColor[0.7, 0.7, 1], LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[0, 0, 1]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] 4 {3, 4}, ViewPoint \[Rule] {1, 0.5, \(-0.3\)}, Background \[Rule] lightblue, Boxed \[Rule] False, ImageSize \[Rule] 1000];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"", "]"}], "\[IndentingNewLine]", \(x[\[Alpha]_, t_] := t {Cos[\[Alpha]], Sin[\[Alpha]], 1};\), "\n", \(co = ParametricPlot3D[ Append[x[\[Alpha], t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {t, \(-3\), 3}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {121, 121}, Axes \[Rule] False, PlotRange \[Rule] All, BoxRatios \[Rule] {1, 1, 1}, Boxed \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(view = {2, 2, 1};\), "\[IndentingNewLine]", \(Show[co, PlotRange \[Rule] All, Axes \[Rule] False, BoxRatios \[Rule] {1, 1, 1}, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] view, ImageSize \[Rule] 1000, DisplayFunction -> $DisplayFunction];\)}], "Input"], Cell[TextData[{ "The following is time-consuming.", StyleBox[" For orientation of the reader, the point (3,0,3) has been made \ visible.", FontVariations->{"CompatibilityType"->0}] }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\n", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\n", \(\(x[\[Alpha]_, t_] := t {Cos[\[Alpha]], Sin[\[Alpha]], 1};\)\), "\[IndentingNewLine]", \(\(raster1 = \[Pi]/100;\)\), "\n", \(\(raster2 = 0.03;\)\), "\[IndentingNewLine]", \(\(point = 0.001;\)\), "\n", \(\(listc = Table[{darkblue, Point[x[\[Alpha], t]]}, {\[Alpha], \(-\[Pi]\) + raster1, \[Pi], raster1}, {t, \(-3\), 3, raster2}];\)\), "\[IndentingNewLine]", \(\(listt = Table[{RGBColor[1, 0, 0], Point[x[\[Alpha], 3]]}, {\[Alpha], \(-\[Pi]\) + raster1, \[Pi], raster1}];\)\), "\[IndentingNewLine]", \(\(listb = Table[{RGBColor[1, 0, 0], Point[x[\[Alpha], \(-3\)]]}, {\[Alpha], \(-\[Pi]\) + raster1, \[Pi], raster1}];\)\), "\[IndentingNewLine]", \(\(po = Graphics3D[{PointSize[12 point], RGBColor[1, 0, 0], Point[x[0, 3]]}, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(co = Graphics3D[{PointSize[point], listc}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(ct = Graphics3D[{PointSize[5 point], listt}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(cb = Graphics3D[{PointSize[5 point], listb}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(view = {2, 2, 1};\)\), "\[IndentingNewLine]", \(\(Show[po, co, ct, cb, PlotRange \[Rule] All, Axes \[Rule] False, BoxRatios \[Rule] {1, 1, 1}, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] view, ImageSize \[Rule] 700, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 4: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 4.2 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["In the first illustration below, ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" adds the two asymptotes without due cause, which therefore are \ removed by hand.", FontVariations->{"CompatibilityType"->0}] }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Eliminate::ifun]\), "\[IndentingNewLine]", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), \ "\[IndentingNewLine]", \(\(<< Graphics`Graphics`;\)\), "\[IndentingNewLine]", \(r[\[Alpha]_] := 6\/\(1 - 2 Cos[\[Alpha]]\)\), "\[IndentingNewLine]", \(\(p1 = PolarPlot[r[\[Alpha]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle -> RGBColor[1, 0, 0], Axes \[Rule] None, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = Show[p1 //. Line[{a___, b : {_, y1_}, c : {_, y2_}, d___}] \[RuleDelayed] Sequence[Line[{a, b}], Line[{c, d}]] /; Abs[y1 - y2] > 10, PlotRange \[Rule] {\(-10\), 10}, DisplayFunction -> Identity];\)\), "\[IndentingNewLine]", \(\(\(Eliminate[{\@\(x\_1\^2 + x\_2\^2\) == 6\/\(1 - 2 Cos[\[Alpha]]\), x\_1\/\@\(x\_1\^2 + x\_2\^2\) == Cos[\[Alpha]]}, Cos[\[Alpha]]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(<< Graphics`ImplicitPlot`\), "\[IndentingNewLine]", \(g[x1_, x2_] := 3 x1\^2 + 24 x1 - x2\^2 + 36\), "\[IndentingNewLine]", \(\(p3 = ImplicitPlot[g[x1, x2] \[Equal] 0, {x1, \(-10.1\), 2.1}, PlotStyle -> RGBColor[1, 0, 0], Axes \[Rule] None, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(Eliminate[{g[x1, x2] \[Equal] 0, x1 == r\ Cos[\[Alpha]], x2 \[Equal] \(-r\)\ \@\(1 - Cos[\[Alpha]]\^2\)}, {x1, x2}]\), "\[IndentingNewLine]", \(Simplify[ g[x1, x2] == 3 \((x1 + 4)\)\^2 - x2\^2 - 12]\), "\[IndentingNewLine]", \(\(\(Simplify[ g[x1, x2] /. Thread[{x1, x2} \[Rule] {\(-4\) + 2 Cosh[t], Sqrt[12] Sinh[t]}]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), \ "\[IndentingNewLine]", \(phi[t_] := {\(-4\) + 2 Cosh[t], 2 \(\@ 3\) Sinh[t]}\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot[phi[t], {t, \(-2\), 2}, AspectRatio \[Rule] Automatic, Axes \[Rule] None, PlotStyle -> RGBColor[1, 0, 0], Background \[Rule] lightblue, ImageSize \[Rule] 200, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(Simplify[ 3 \( phi[t]\)[\([1]\)]\^2 + 24 \( phi[t]\)[\([1]\)] - \(phi[t]\)[\([2]\)]\^2 + 36]\), "\[IndentingNewLine]", \(Eliminate[{x\_1 + 4 == 2 Cosh[t], x\_2 == \(\@12\) Sinh[t]}, t]\), "\[IndentingNewLine]", \(Eliminate[{x1 + 4 == 2 Cosh[t], x2 == \(\@12\) Sinh[t], x1 \[Equal] r\ Cos[\[Alpha]], x2 \[Equal] r\ Sin[\[Alpha]]}, {x1, x2, t}]\), "\[IndentingNewLine]", \(\(\(\(Solve[ Simplify[% /. Sin[\[Alpha]]\^2 \[Rule] 1 - Cos[\[Alpha]]\^2], r]\)[\([1]\)] /. {Rule \[Rule] Equal}\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(Show[GraphicsArray[{p2, p3, p4}], ImageSize \[Rule] 1200, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.4 (Cycloid) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Solve::tdep]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Phi][t_] := {t - Sin[t], 1 - Cos[t]}\), "\[IndentingNewLine]", \(Solve[\[Phi][t] \[Equal] \[Phi][u], {t, u}]\), "\[IndentingNewLine]", \(Reduce[\(\[Phi][t]\)[\([2]\)] \[Equal] \(\[Phi][u]\)[\([2]\)], t]\), "\[IndentingNewLine]", \(Simplify[%[\([2]\)], {0 < u < \[Pi]}]\ \), "\[IndentingNewLine]", \(Solve[{\(\[Phi][t]\)[\([1]\)] \[Equal] \(\[Phi][ t + 2 \[Pi]]\)[\([1]\)]}, t]\), "\[IndentingNewLine]", \(Solve[{\(\[Phi][t]\)[\([1]\)] \[Equal] \(\[Phi][ t - 2 \[Pi]]\)[\([1]\)]}, t]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.5 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(amin = \(-\[Pi]\);\)\), "\[IndentingNewLine]", \(\(amax = \[Pi];\)\), "\[IndentingNewLine]", \(\(tmin = \(-\[Pi]\);\)\), "\[IndentingNewLine]", \(\(tmax = \[Pi];\)\), "\[IndentingNewLine]", \(\(na = 101;\)\), "\[IndentingNewLine]", \(\(nt = 301;\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]]}\), "\[IndentingNewLine]", \(MatrixForm[\[Phi][\[Alpha], \[Theta]]]\), "\[IndentingNewLine]", \(Simplify[ Eliminate[{x\_1 == \(\[Phi][\[Alpha], \[Theta]]\)[\([1]\)], x\_2 == \(\[Phi][\[Alpha], \[Theta]]\)[\([2]\)], x\_3 == \(\[Phi][\[Alpha], \[Theta]]\)[\([3]\)]}, {\[Alpha], \ \[Theta]}]]\), "\[IndentingNewLine]", \(\(\(ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]]] // Evaluate, {\[Alpha], amin, amax}, {\[Theta], tmin, tmax}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {na, nt}, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(-1.1\), 1.1}, {\(-1.1\), 1.1}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(\(\((j = Transpose[{\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]], \ \[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]]}])\) // MatrixForm\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(pt = Table[ParametricPlot3D[\[Phi][\[Alpha], \[Theta]] // Evaluate, {\[Alpha], amin, amax}, DefaultColor \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, PlotPoints \[Rule] nt, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(-1.1\), 1.1}, {\(-1.1\), 1.1}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, DisplayFunction \[Rule] Identity], {\[Theta], tmin, tmax, \[Pi]/80}];\)\), "\[IndentingNewLine]", \(\(\(Show[pt, ImageSize \[Rule] 1000, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(pa = Table[ParametricPlot3D[\[Phi][\[Alpha], \[Theta]] // Evaluate, {\[Theta], tmin, tmax}, DefaultColor \[Rule] darkblue, Background \[Rule] lightblue, PlotPoints \[Rule] nt, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(-1.1\), 1.1}, {\(-1.1\), 1.1}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, DisplayFunction \[Rule] Identity], {\[Alpha], amin, amax, \[Pi]/80}];\)\), "\[IndentingNewLine]", \(\(Show[pa, ImageSize \[Rule] 1000, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.6 (Surfaces of revolution) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(Off[Eliminate::ifun]\), "\[IndentingNewLine]", \(\(smin = \(-3\);\)\), "\[IndentingNewLine]", \(\(smax = 3;\)\), "\[IndentingNewLine]", \(\(tmin = \(-\[Pi]\);\)\), "\[IndentingNewLine]", \(\(tmax = \[Pi];\)\), "\[IndentingNewLine]", \(\(ns = 101;\)\), "\[IndentingNewLine]", \(\(nt = 301;\)\), "\[IndentingNewLine]", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print[Catenary]\), "\[IndentingNewLine]", \(\[Gamma][s_] := {Cosh[s], 0, s}\), "\[IndentingNewLine]", \(\(\(ParametricPlot3D[\[Gamma][s], {s, smin, smax}, DefaultColor \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, PlotPoints \[Rule] ns, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-2\), 2}, {\(-2\), 2}, {\(-1.2\), 1.2}}, Boxed \[Rule] True, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 600];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print[Catenoid]\), "\[IndentingNewLine]", \(\[Phi][s_, t_] := {\(\[Gamma][s]\)[\([1]\)] Cos[t], \(\[Gamma][s]\)[\([1]\)] Sin[t], \(\[Gamma][s]\)[\([3]\)]}\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[\[Phi][s, t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, smin, smax}, {t, tmin, tmax}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {ns, nt}, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-2\), 2}, {\(-2\), 2}, {\(-1.2\), 1.2}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\ \[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Gamma][s_] := {\[Gamma]\_1[s], 0, \[Gamma]\_3[s]}\), "\[IndentingNewLine]", \(\[Phi][s, t] // MatrixForm\), "\[IndentingNewLine]", \(\((j = Transpose[{\[PartialD]\_s\ \[Phi][s, t], \[PartialD]\_t\ \[Phi][s, t]}])\) // MatrixForm\), "\[IndentingNewLine]", \(\(\(FullSimplify[\(-Sin[t]\) j[\([1, 2]\)] + Cos[t] j[\([2, 2]\)] \[Equal] 0]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Gamma][s_] := {s\^3, 0, s}\), "\[IndentingNewLine]", \(\(ParametricPlot3D[\[Gamma][s], {s, smin, smax}, DefaultColor \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, PlotPoints \[Rule] ns, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-2\), 2}, {\(-2\), 2}, {\(-1.2\), 1.2}}, Boxed \[Rule] True, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[\[Phi][s, t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, smin, smax}, {t, tmin, tmax}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {ns, nt}, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-2\), 2}, {\(-2\), 2}, {\(-1.2\), 1.2}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\ \), "\[IndentingNewLine]", \(\[Gamma][s_] := {\@s, 0, s}\), "\[IndentingNewLine]", \(\(ParametricPlot3D[\[Gamma][s], {s, 0, smax}, DefaultColor \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, PlotPoints \[Rule] ns, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-2\), 2}, {\(-2\), 2}, {0, 2}}, Boxed \[Rule] True, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[\[Phi][s, t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, 0, smax}, {t, tmin, tmax}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {ns, nt}, ViewPoint \[Rule] {1, 2, 0.8}, PlotRange \[Rule] {{\(-2\), 2}, {\(-2\), 2}, {0, 2}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\ \[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Eliminate[{x1 \[Equal] a\ Cosh[s] Cos[t], x2 \[Equal] a\ Cosh[s] Sin[t], x3 \[Equal] a\ s}, {s, t}] /. Thread[{x1, x2, x3} \[Rule] {x\_1, x\_2, x\_3}]\), "\[IndentingNewLine]", \(\(\(Eliminate[{x1 \[Equal] a\ Cosh[s] Cos[t], x2 \[Equal] a\ Cosh[s] Sin[t], x3 \[Equal] a\ s}, {s, t}]\)[\([6]\)] /. Thread[{x1, x2, x3} \[Rule] {x\_1, x\_2, x\_3}];\)\), "\[IndentingNewLine]", \(\((% /. {s \[Rule] x3/a})\) /. Thread[{x1, x2, x3} \[Rule] {x\_1, x\_2, x\_3}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.8 (Helix and helicoid) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(helicoid[s_, t_] := {t\ Cos[s], t\ Sin[s], s}\), "\n", \(\(ParametricPlot3D[ Append[helicoid[s, t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, 0, \(5 \[Pi]\)\/2}, {t, \(-6\), 6}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Axes \[Rule] None, Boxed \[Rule] False, ViewPoint \[Rule] {1, \(-2\), 0.8}, PlotPoints \[Rule] {55, 45}, Background \[Rule] lightblue, ImageSize \[Rule] 800];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.11 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(Off[Solve::svars]\), "\[IndentingNewLine]", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(g[x1_, x2_] := x1\^3 - x2\^3\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(Plot[g[x1, 0], {x1, \(-3\), 3}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotRange \[Rule] All, ImageSize \[Rule] 700];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(j = {{\[PartialD]\_\(x\_1\)\ g[x\_1, x\_2], \[PartialD]\_\(x\_2\)\ g[x\_1, x\_2]}};\)\ \), "\[IndentingNewLine]", \(MatrixForm[j]\), "\[IndentingNewLine]", \(First[ Solve[j \[Equal] 0, {x\_1, x\_2}]] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(First[ Solve[g[x\_1, x\_2] \[Equal] c, {x\_1, x\_2}]]\), "\[IndentingNewLine]", \(<< Graphics`ImplicitPlot`\), "\[IndentingNewLine]", \(\(p := Table[ImplicitPlot[g[x1, x2] \[Equal] c, {x1, \(-4\), 4}, Axes \[Rule] None, Ticks \[Rule] None, PlotStyle -> RGBColor[\((c + 3)\)/6, 0, 1 - \((c + 3)\)/6], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity], {c, \(-3\), 3, 1/4}];\)\), "\[IndentingNewLine]", \(\(Show[p, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.12 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(g[ x1_, x2_, x3_] := x1\^2 + x2\^2 - x3\^2\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(p1 = Plot[g[x1, 0, 0], {x1, 0, 10}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(p2 = Plot[g[0, 0, x3], {x3, 0, 10}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(Show[ GraphicsArray[{p1, p2}], ImageSize \[Rule] 1200, DisplayFunction -> $DisplayFunction];\), "\[IndentingNewLine]", \ \(Print["\"]\), "\[IndentingNewLine]", RowBox[{\(Flatten[ Solve[{{\[PartialD]\_\(x\_1\)\ g[x\_1, x\_2, x\_3], \[PartialD]\_\(x\_2\)\ g[x\_1, x\_2, x\_3], \[PartialD]\_\(x\_3\)\ g[x\_1, x\_2, x\_3]} \[Equal] {0, 0, 0}}, {x\_1, x\_2, x\_3}]] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"", "]"}], "\n", \(x[\[Alpha]_, t_] := t {Cos[\[Alpha]], \ Sin[\[Alpha]], 1};\), "\[IndentingNewLine]", \(co = ParametricPlot3D[ x[\[Alpha], t], {\[Alpha], \(-\[Pi]\), \[Pi]}, {t, \(-3\), 3}, PlotPoints \[Rule] {121, 121}, Axes \[Rule] False, PlotRange \[Rule] All, BoxRatios \[Rule] {1, 1, 1}, Boxed \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(view = {2, 2, 1};\), "\[IndentingNewLine]", \(Show[co, PlotRange \[Rule] All, Axes \[Rule] False, BoxRatios \[Rule] {1, 1, 1}, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] view, ImageSize \[Rule] 700, DisplayFunction -> $DisplayFunction];\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.14 (Quadrics) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ "See also the file Ex4.14.pdf. Here a representative example for the case \ of ", StyleBox["n", FontSlant->"Italic"], "=3 is given. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\((A = {{2, \(-1\), 0}, {\(-1\), 2, \(-1\)}, {0, \(-1\), 2}})\) // MatrixForm\ \), "\[IndentingNewLine]", \(Eigenvalues[A]\), "\[IndentingNewLine]", \(\(b = {\(-1\), 1, 1};\)\), "\[IndentingNewLine]", \(co[x_, n_] := Table[x\_j, {j, n}]\), "\[IndentingNewLine]", \(\(xv = co[x, 3];\)\), "\[IndentingNewLine]", \(g[xv_] := \((A . xv)\) . xv + b . xv + c\), "\[IndentingNewLine]", \(Expand[g[xv]]\), "\[IndentingNewLine]", \(\[CapitalDelta] = \((Inverse[A] . b)\) . b - 4 c\), "\[IndentingNewLine]", \(\(\(v = \(-\(1\/2\)\) Inverse[A] . b\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(c = 1\/2;\)\), "\[IndentingNewLine]", \(\[CapitalDelta]\), "\[IndentingNewLine]", \(g[v]\), "\[IndentingNewLine]", \(Flatten[ Solve[{\[PartialD]\_\(x\_1\)\ g[xv], \[PartialD]\_\(x\_2\)\ g[xv], \[PartialD]\_\(x\_3\)\ g[xv]} \[Equal] {0, 0, 0}, xv]] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]", \(\(yv = co[y, 3];\)\), "\[IndentingNewLine]", \(xv = yv + v\), "\[IndentingNewLine]", \(Simplify[g[xv]]\), "\[IndentingNewLine]", \(2 \((\((yv[\([1]\)] - yv[\([2]\)]\/2)\)\^2 + yv[\([2]\)]\^2\/2 + \((\(-\(yv[\([2]\)]\/2\)\) + yv[\([3]\)])\)\^2)\ \)\), "\[IndentingNewLine]", \(Simplify[ 2 \((\((yv[\([1]\)] - yv[\([2]\)]\/2)\)\^2 + yv[\([2]\)]\^2\/2 + \((\(-\(yv[\([2]\)]\/2\)\) + \ yv[\([3]\)])\)\^2)\)]\), "\[IndentingNewLine]", \(\(\(Flatten[ Solve[{yv[\([1]\)] - yv[\([2]\)]\/2 \[Equal] 0, yv[\([2]\)] \[Equal] 0, \(-\(yv[\([2]\)]\/2\)\) + yv[\([3]\)] \[Equal] 0}, {yv[\([1]\)], yv[\([2]\)], yv[\([3]\)]}]] /. {Rule \[Rule] Equal}\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\ 0\>"]\), "\[IndentingNewLine]", \(\(c = 0;\)\), "\[IndentingNewLine]", \(\[CapitalDelta] = \((Inverse[A] . b)\) . b - 4 c\), "\[IndentingNewLine]", \(g[v]\), "\[IndentingNewLine]", \(Expand[g[xv]]\), "\[IndentingNewLine]", \(Simplify[ 2 \((\((yv[\([1]\)] - yv[\([2]\)]\/2)\)\^2 + yv[\([2]\)]\^2\/2 + \((\(-\(yv[\([2]\)]\/2\)\) + yv[\([3]\)])\)\ \^2)\) - 1\/2]\), "\[IndentingNewLine]", \(\((yv[\([1]\)] - yv[\([2]\)]\/2)\)\^2 + yv[\([2]\)]\^2\/2 + \((\(-\(yv[\([2]\)]\/2\)\) + yv[\([3]\)])\)\^2 == 1\/4\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.17 (Lines on hyperboloid of one sheet) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\n", \(\ \(hplus[a_, b_, c_]\)[s_, t_] := {a\ Cos[s] - a\ t\ Sin[s], b\ t\ Cos[s] + b\ Sin[s], c\ t}\), "\n", \(\(hminus[a_, b_, c_]\)[s_, t_] := {a\ Cos[s] + a\ t\ Sin[s], \(-b\)\ t\ Cos[s] + b\ Sin[s], c\ t}\), "\n", \(\(p1 = ParametricPlot3D[\ \ \(hplus[1, 2, 3]\)[s, t] // Evaluate, {s, 0, 2 \[Pi]}, {t, \(-1.5\), 1.5}, Axes \[Rule] None, Boxed \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = ParametricPlot3D[\(hminus[1, 2, 3]\)[s, t] // Evaluate, {s, 0, 2 \[Pi]}, {t, \(-1.5\), 1.5}, Axes \[Rule] None, Boxed \[Rule] False, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[GraphicsArray[{p1, p2}, GraphicsSpacing \[Rule] 0], ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Exercise 4.22 (Rotations in ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "3"], TraditionalForm]]], " and Euler's formula) " }], "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(<< Calculus`VectorAnalysis`\), "\[IndentingNewLine]", \(<< LinearAlgebra`Orthogonalization`\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\n", \(\(\[Alpha]0 = \[Pi]/6;\)\), "\n", \(\(b = {\(-0.2\), 0.7, 1.5};\)\), "\n", \(\(a = Normalize[b];\)\), "\[IndentingNewLine]", \(cp[a_, x_] := \ CrossProduct[a, x]\), "\[IndentingNewLine]", \(\ \ \(s[y_] := Cos[\[Alpha]0]\ y + Sin[\[Alpha]0] cp[a, x];\)\), "\n", \(\ \ \(r[x_] := \((1 - Cos[\[Alpha]0])\) \((a . x)\)\ a + Cos[\[Alpha]0]\ x + Sin[\[Alpha]0]\ cp[a, x];\)\), "\n", \(\(xx[t_] := \((1 - Cos[t])\)\ \((a . x)\)\ a + Cos[t]\ x + Sin[t]\ cp[a, x];\)\), "\n", \(\(yy[t_] := Cos[t]\ y + Sin[t]\ cp[a, x];\)\), "\n", \(\(x0 = {\(-0.6\), \(-0.1\), 1};\)\), "\n", \(\(x = Normalize[x0];\)\), "\n", \(\(y = x - \((x . a)\)\ a;\)\), "\[IndentingNewLine]", \(\(p1 = Graphics3D[{darkblue, AbsolutePointSize[6], {Point[a], Point[x], Point[\((x . a)\)\ a], Point[y], Point[s[y]], Point[r[x]], Point[{0, 0, 0}], Point[cp[a, x]], Line[{a, {0, 0, 0}, y, x, \((x . a)\)\ a, r[x], s[y], Cos[\[Alpha]0]\ y, {0, 0, 0}, cp[a, x]}], Line[{{0, 0, 0}, s[y], Sin[\[Alpha]0]\ cp[a, x]}]}, Line[{{0, 0, 0}, x}], Line[{a, {0, 0, 0}}]}];\)\), "\n", \(\(p2 = Graphics3D[{RGBColor[1, 0, 0], Line[{x, {0, 0, 0}}]}];\)\), "\[IndentingNewLine]", \(\(p3 = Graphics3D[{darkblue, {Line[{x - 0.85\ \((x - \((x . a)\)\ a)\), x - 0.85\ \((x - \((x . a)\)\ a)\) - 0.125\ \((x . a)\)\ a, 0.875\ \((x . a)\)\ a}], Line[{0.1\ \((a . x)\)\ a, 0.1\ \((a . x)\)\ a + 0.15\ cp[a, x], 0.15\ cp[a, x]}], Line[{0.15\ Cos[\[Alpha]0]\ y, 0.15\ Cos[\[Alpha]0]\ y + 0.1\ \((a . x)\)\ a, 0.1\ \((a . x)\)\ a}], Line[{0.15\ Cos[\[Alpha]0]\ y, 0.15\ Cos[\[Alpha]0]\ y + 0.15\ cp[a, x], 0.15\ cp[a, x]}]}}];\)\), "\n", \(\(p4 = Graphics3D[{Text["\", a], Text["\", x], Text["\", y], Text["\", r[x]], Text["\", s[y]], Text["\", Cos[\[Alpha]0]\ y], Text["\", cp[a, x]], Text["\", \((x . a)\)\ a], Text["\", 0.5\ cp[a, x]], Text["\", 0.5\ s[y]]}];\)\), "\n", \(\(p5 = Show[p1, p2, p3, p4, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] {1, \(-3\), 4}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p6 = ParametricPlot3D[Append[xx[t], RGBColor[1, 0, 0]], {t, 0, \[Alpha]0}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p7 = ParametricPlot3D[Append[yy[t], RGBColor[1, 0, 0]], {t, 0, \[Pi]/2}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p8 = ParametricPlot3D[Append[0.4\ yy[t], darkblue], {t, 0, \[Alpha]0}, PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[p5, p6, p7, p8, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] {1, \(-8\), 8}, DefaultFont \[Rule] 12. , ImageSize \[Rule] 700, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"], Cell[TextData[{ StyleBox["See also Exercises 2.5 and 5.58 on the exponential of an \ antisymmetric matrix. In these exercises \[Alpha] is a real number and ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ RowBox[{"(", RowBox[{ SubscriptBox[ StyleBox["a", FontSlant->"Italic"], "1"], ",", SubscriptBox[ StyleBox["a", FontSlant->"Italic"], "2"], ",", SubscriptBox[ StyleBox["a", FontSlant->"Italic"], "3"]}], ")"}]]], StyleBox[" is a unit vector in ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "3"], TraditionalForm]]], StyleBox[". ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]] }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[ General::spell1]\), "\n", \(co[x_, n_] := Table[x\_j, {j, n}]\), "\n", \(Unprotect[Power];\), "\n", RowBox[{\(Format[Power[Subscript[x_, y_], z_]] := DisplayForm[SuperscriptBox[SubscriptBox[x, y], z]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(angle[a_List] := ArcCos[\(\(-1\) + a[\([1, 1]\)] + a[\([2, 2]\)] + a[\([3, \ 3]\)]\)\/2]\), "\[IndentingNewLine]", \(ax[a_List] := Simplify[\(a - Transpose[a]\)\/\(2 Sin[angle[a]]\)]\), "\ \[IndentingNewLine]", RowBox[{\(axis[ a_List] := {\(ax[a]\)[\([3, 2]\)], \(-\(ax[a]\)[\([3, 1]\)]\), \(ax[ a]\)[\([2, 1]\)]}\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(mat[a_, p_, n_] := Table[Subscript[a, ToString[j] <> ToString[i]], {i, p}, {j, n}];\), "\[IndentingNewLine]", \(MatrixForm[ mat[a, 3, 3]]\), "\[IndentingNewLine]", \(angle[ mat[a, 3, 3]]\), "\[IndentingNewLine]", RowBox[{\(axis[mat[a, 3, 3]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\)\_1\),\!\(\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\_2\),\!\(\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\_3\))\>\"", "]"}], "\[IndentingNewLine]", \(r[{a1_, a2_, a3_}] := {{0, \(-a3\), a2}, {a3, 0, \(-a1\)}, {\(-a2\), a1, 0}};\), "\n", RowBox[{\(R[an_, a_List] := Cos[an] IdentityMatrix[3] + \((1 - Cos[an])\) Transpose[{a}] . {a} + Sin[an] r[a];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(av = co[a, 3];\), "\[IndentingNewLine]", \(MatrixForm[ r[av]]\), "\[IndentingNewLine]", RowBox[{ "Print", "[", "\"\\"Italic\"]\)\_1\),\!\(\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\_2\),\!\(\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\_3\))\>\"", "]"}], "\[IndentingNewLine]", RowBox[{\(MatrixForm[R[\[Alpha], av]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", RowBox[{\(Simplify[ Transpose[R[\[Alpha], av]] . R[\[Alpha], av] \[Equal] IdentityMatrix[3]] /. av[\([1]\)]\^2 -> 1 - av[\([2]\)]\^2 - av[\([3]\)]\^2\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ "Print", "[", "\"\\"Italic\"]\)\_1\),\!\(\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\_2\),\!\(\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\_3\))\>\"", "]"}], "\[IndentingNewLine]", \(MatrixForm[ ComplexExpand[MatrixExp[\[Alpha]\ r[av]]] /. av[\([1]\)]\^2 -> 1 - av[\([2]\)]\^2 - av[\([3]\)]\^2]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Simplify[ ComplexExpand[MatrixExp[\[Alpha]\ r[av]] \[Equal] R[\[Alpha], av]] /. av[\([1]\)]\^2 -> 1 - av[\([2]\)]\^2 - av[\([3]\)]\^2]\)}], "Input"], Cell[TextData[StyleBox["As a verification, we compute the example from the \ exercise; next several orthogonal matrices of determinant 1 and their \ corresponding angles and axes of rotation; and finally some matrix \ exponentials.", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]]], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Print["\"]\), "\[IndentingNewLine]", \(\((rot = R[\[Pi]\/2, {1, 0, 0}] . R[\[Pi]\/2, {0, 1, 0}])\) // MatrixForm\), "\[IndentingNewLine]", \(MatrixForm[Transpose[rot] . rot]\), "\[IndentingNewLine]", \(angle[rot]\), "\[IndentingNewLine]", \(\(\@3\) axis[rot]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\((rot = Simplify[R[\[Pi]\/3, {\@\(1\/5\), \@\(2\/5\), \@\(2\/5\)}]])\) // MatrixForm\), "\[IndentingNewLine]", \(MatrixForm[Together[Transpose[rot] . rot]]\), "\[IndentingNewLine]", \(angle[rot]\), "\[IndentingNewLine]", \(axis[rot]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\((rot = Simplify[ R[\(2 \[Pi]\)\/3, {\(-\(3\/4\)\), \(-\(1\/4\)\), \@6\/4}]])\) // MatrixForm\), "\[IndentingNewLine]", \(MatrixForm[Together[Transpose[rot] . rot]]\), "\[IndentingNewLine]", \(angle[rot]\), "\[IndentingNewLine]", \(axis[rot]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), \ "\[IndentingNewLine]", \(MatrixForm[ MatrixExp[\[Alpha]\ {{0, 0, 0}, {0, 0, \(-1\)}, {0, 1, 0}}]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), \ "\[IndentingNewLine]", \(MatrixForm[ MatrixExp[\(2 \[Pi]\)\/3\ {{0, \(-\(\@6\/4\)\), \(-\(1\/4\)\)}, \ {\@6\/4, 0, 3\/4}, {1\/4, \(-\(3\/4\)\), 0}}]]\), "\[IndentingNewLine]", \(MatrixForm[Simplify[% - rot]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 4.26 (Hopf fibration and stereographic projection) \ \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(na = 3;\)\), "\[IndentingNewLine]", \(\(nb = 40;\)\), "\[IndentingNewLine]", \(\(nu = 500;\)\), "\[IndentingNewLine]", \(\(point = 0.002;\)\), "\[IndentingNewLine]", \(\(viewx = \(-2\);\)\), "\[IndentingNewLine]", \(\(viewy = \(-2\);\)\), "\[IndentingNewLine]", \(\(viewz = 0.7;\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(hopf[a_, b_, u_] := {Cos[a]\ Cos[u], Cos[a]\ Sin[u], Sin[a]\ Cos[\((u + b)\)], Sin[a]\ Sin[\((u + b)\)]}\), "\n", \(sterproj[{x_, y_, z_, t_}] := {x/\((1 - t)\), y/\((1 - t)\), z/\((1 - t)\)}\), "\[IndentingNewLine]", \(\(list = Table[{darkblue, Point[sterproj[hopf[a, b, u]]]}, {a, 0, \[Pi], \[Pi]/\((2\ na + 1)\)}, {b, 0, 2 \[Pi], 2\ \[Pi]/nb}, {u, 0, 2 \[Pi], 2\ \[Pi]/nu}];\)\), "\[IndentingNewLine]", \(\(Show[ Graphics3D[{PointSize[point], list}, Axes \[Rule] False, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] {viewx, viewy, viewz}], Background \[Rule] lightblue, ImageSize \[Rule] 1100];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.27 (Steiner's Roman surface) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\[Alpha]min = 0;\)\), "\[IndentingNewLine]", \(\(\[Alpha]max = 2 \[Pi];\)\), "\[IndentingNewLine]", \(\(\[Theta]min = \(-\[Pi]\)/2;\)\), "\[IndentingNewLine]", \(\(\[Theta]max = \[Pi]/2;\)\), "\[IndentingNewLine]", \(\(n\[Alpha] = 101;\)\), "\[IndentingNewLine]", \(\(n\[Theta] = 37;\)\), "\[IndentingNewLine]", \(\(y1[\[Alpha]_, \[Theta]_] := Cos[\[Alpha]] Cos[\[Theta]];\)\), "\[IndentingNewLine]", \(\(y2[\[Alpha]_, \[Theta]_] := Sin[\[Alpha]] Cos[\[Theta]];\)\), "\[IndentingNewLine]", \(\(y3[\[Alpha]_, \[Theta]_] := Sin[\[Theta]];\)\), "\[IndentingNewLine]", \(\[CapitalPsi][y1_, y2_, y3_] := {y2\ y3, y3\ y1, y1\ y2}\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := \[CapitalPsi][y1[\[Alpha], \[Theta]], y2[\[Alpha], \[Theta]], y3[\[Alpha], \[Theta]]]\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \[Alpha]min, \[Alpha]max}, {\[Theta], \ \[Theta]min, \[Theta]max}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {3 n\[Alpha], 3 n\[Theta]}, ViewPoint \[Rule] {1, 2, 2}, PlotRange \[Rule] {{\(-0.5\), 0.5}, {\(-0.5\), 0.5}, {\(-0.5\), 0.5}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \[Alpha]min, \[Alpha]max}, {\[Theta], \ \[Theta]min, \[Theta]max}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {3 n\[Alpha], 3 n\[Theta]}, ViewPoint \[Rule] {1, 2, 2}, PlotRange \[Rule] {{\(-0.5\), 0.5}, {\(-0.5\), 0.5}, {\(-0.5\), 0.25}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \[Alpha]min, \[Alpha]max}, {\[Theta], \ \[Theta]min, \[Theta]max}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {3 n\[Alpha], 3 n\[Theta]}, ViewPoint \[Rule] {1, 2, 2}, PlotRange \[Rule] {{\(-0.5\), 0.15}, {\(-0.5\), 0.5}, {\(-0.5\), 0.5}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\)}], "Input"], Cell[TextData[{ "The following is time-consuming.\n\n", StyleBox["In order to start the movie, double click in the plot on the \ screen; and for stopping click once.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 0.500008], FontVariations->{"CompatibilityType"->0}] }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\[Alpha]min = 0;\)\), "\[IndentingNewLine]", \(\(\[Alpha]max = 2 \[Pi];\)\), "\[IndentingNewLine]", \(\(\[Theta]min = \(-\[Pi]\)/2;\)\), "\[IndentingNewLine]", \(\(\[Theta]max = \[Pi]/2;\)\), "\[IndentingNewLine]", \(\(n\[Alpha] = 101;\)\), "\[IndentingNewLine]", \(\(n\[Theta] = 37;\)\), "\[IndentingNewLine]", \(\(y1[\[Alpha]_, \[Theta]_] := Cos[\[Alpha]] Cos[\[Theta]];\)\), "\[IndentingNewLine]", \(\(y2[\[Alpha]_, \[Theta]_] := Sin[\[Alpha]] Cos[\[Theta]];\)\), "\[IndentingNewLine]", \(\(y3[\[Alpha]_, \[Theta]_] := Sin[\[Theta]];\)\), "\[IndentingNewLine]", \(\[CapitalPsi][y1_, y2_, y3_] := {y2\ y3, y3\ y1, y1\ y2}\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := \[CapitalPsi][y1[\[Alpha], \[Theta]], y2[\[Alpha], \[Theta]], y3[\[Alpha], \[Theta]]]\), "\[IndentingNewLine]", \(\(p[n_] := ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \[Alpha]min, \[Alpha]max}, {\[Theta], \ \[Theta]min, \[Theta]max}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, PlotPoints \[Rule] {3 n\[Alpha], 3 n\[Theta]}, ViewPoint \[Rule] {1, 2, 2}, PlotRange \[Rule] {{\(-0.5\), 0.5}, {\(-0.5\), 0.5}, {\(-0.5\), \(-0.5\) + n/12\ 0.5}}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\), "\[IndentingNewLine]", \(\(SetOptions[SelectedNotebook[], AnimationDisplayTime -> .2];\)\ \ \ \ \ \ \ \ \), "\n", \(\(SetOptions[SelectedNotebook[], CellGrouping -> Automatic];\)\ \ \ \ \ \ \), "\[IndentingNewLine]", \(\(movie := Table[Show[p[n], ImageSize \[Rule] 1000, DisplayFunction \[Rule] $DisplayFunction], {n, 0, 24}];\)\), "\[IndentingNewLine]", \(\(movie;\)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \), "\n", \(\(SelectionMove[SelectedNotebook[], All, GeneratedCell];\)\), "\n", \(\(FrontEndExecute[FrontEndToken["\"]];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.28 (Cayley's surface) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[ General::spell]\), "\n", \(Off[ General::spell1]\), "\n", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", RowBox[{\(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(x1min = \(-0.5\);\), \ "\n", \(x1max = 2.5;\), "\n", \(x2max = 1.5;\), "\n", \(x2min = \(-x2max\);\), "\n", \(x2maxc = Min[x1max, Sqrt[x1max/ 3]];\), "\n", \(x2minc = \(-x2maxc\);\), "\n", \(planex2 = 6;\), "\[IndentingNewLine]", \(planex3 = \(-6\);\), "\n", \(viewx1 = 1;\), "\n", \(viewx2 = \(-3\);\), "\n", \(viewx3 = 2.5;\), "\[IndentingNewLine]", \(raster = 0.05;\), "\n", \(point = 0.001;\), "\n", RowBox[{\(dash = 0.01;\), "\[IndentingNewLine]"}], "\n", \(f[x1_, x2_] := x2^3 - x1\ x2\), "\n", \(h[x2_] := 3 x2^2\), "\[IndentingNewLine]", \(list1 = Table[{darkblue, Point[{x1, x2, f[x1, x2]}]}, {x1, x1min, x1max, raster}, {x2, x2min, x2max, raster}];\), "\n", \(list2 = Table[{darkblue, Point[{x1, x2, planex3}]}, {x1, x1min, x1max, raster}, {x2, x2min, x2max, raster}];\), "\n", \(list3 = Table[{darkblue, Point[{x1, planex2, f[x1, x2]}]}, {x1, x1min, x1max, raster}, {x2, x2min, x2max, raster}];\), "\n", \(cloud = Graphics3D[{PointSize[point], list1, list2, list3}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] {viewx1, viewx2, viewx3}, DisplayFunction \[Rule] Identity];\), "\n", \(x1boundaries = ParametricPlot3D[{{x1, x2min, f[x1, x2min]}, {x1, x2max, f[x1, x2max]}, {x1, x2min, planex3}, {x1, x2max, planex3}, {x1, planex2, f[x1, x2min]}, {x1, planex2, f[x1, x2max]}}, {x1, x1min, x1max}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, ViewPoint \[Rule] {viewx1, viewx2, viewx3}, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\), "\n", \(x2boundaries = ParametricPlot3D[{{x1min, x2, f[x1min, x2]}, {x1max, x2, f[x1max, x2]}, {x1min, x2, planex3}, {x1max, x2, planex3}, {x1min, planex2, f[x1min, x2]}, {x1max, planex2, f[x1max, x2]}}, {x2, x2min, x2max}, Axes \[Rule] False, PlotRange \[Rule] All, ViewPoint \[Rule] {viewx1, viewx2, viewx3}, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\), "\n", \(crit = ParametricPlot3D[{{h[x2], x2, f[h[x2], x2]}, {h[x2], x2, planex3}, {h[x2], planex2, f[h[x2], x2]}}, {x2, x2minc, x2maxc}, Axes \[Rule] False, ViewPoint \[Rule] {viewx1, viewx2, viewx3}, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\), "\n", \(planes = Graphics3D[{Dashing[{dash, dash}], Line[{{x1min, x2min, planex3}, {x1min, planex2, planex3}}], Line[{{x1max, x2min, planex3}, {x1max, planex2, planex3}}], Line[{{x1min, planex2, planex3}, {x1min, planex2, f[x1min, x2min]}}], Line[{{x1max, planex2, planex3}, {x1max, planex2, f[x1max, x2min]}}], Line[{{x1min, planex2, planex3}, {x1max, planex2, planex3}}]}, Axes \[Rule] False, ViewPoint \[Rule] {viewx1, viewx2, viewx3}, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\), "\n", RowBox[{\(Show[cloud, x1boundaries, x2boundaries, crit, planes, DefaultColor \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\!\(\* StyleBox[\"g\",\nFontSlant->\"Italic\"]\) is submersive, as follows from\>\"", "]"}], "\[IndentingNewLine]", \(g[{x1_, x2_, x3_}] := x2^3 - x1\ x2 - x3\), "\[IndentingNewLine]", RowBox[{\(Flatten[ Solve[{{\[PartialD]\_\(x\_1\)g[{x\_1, x\_2, x\_3}], \[PartialD]\_\(x\_2\)g[{x\_1, x\_2, x\_3}], \[PartialD]\_\(x\_3\)g[{x\_1, x\_2, x\_3}]} \[Equal] {0, 0, 0}}, {x\_1, x\_2, x\_3}]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[CurlyPhi][{y1_, y2_, y3_}] := {y1, y2, y2^3 - y1\ y2}\), "\[IndentingNewLine]", \(\(Composition[ g, \[CurlyPhi]]\)[{y1, y2, y3}]\), "\[IndentingNewLine]", RowBox[{\(Flatten[ Solve[g[{y\_1, y\_2, y\_3}] \[Equal] 0, y\_3]] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Pi]\[Pi][{x1_, x2_, x3_}] := {x1, 0, x3}\), "\[IndentingNewLine]", \(\[CapitalXi] = Composition[\[Pi]\[Pi], \[CurlyPhi]];\), "\[IndentingNewLine]", \(\ \[CapitalXi][{x\_1, x\_2, 0}]\), "\[IndentingNewLine]", \ \(\[PartialD]\_\(x\_1\)\[CapitalXi][{x\_1, x\_2, 0}] \[Equal] {0, 0, 0}\), "\[IndentingNewLine]", RowBox[{\(\[PartialD]\_\(x\_2\)\[CapitalXi][{x\_1, x\_2, 0}] \[Equal] {0, 0, 0}\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", RowBox[{\(Simplify[ Eliminate[\[CapitalXi][{3 x\_2^2, x\_2, 0}] \[Equal] {y\_1, 0, y\_3}, x\_2]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(h[x1_, x3_] := 4 x1^3 - 27 x3^2\), "\[IndentingNewLine]", \(First[ Solve[{{\[PartialD]\_\(x\_1\)h[x\_1, x\_3], \[PartialD]\_\(x\_3\)h[ x\_1, x\_3]} \[Equal] {0, 0}}, {x\_1, x\_3}]] /. {Rule \[Rule] Equal}\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 4.29 (Whitney's umbrella) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[{y1\ y2, y1, y2\ y2}, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {y1, \(-1\), 1}, {y2, \(-1\), 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotPoints \[Rule] {51, 55}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Graphics3D[{Text["\", {1.6, 0, 0}], Text["\", {0, 1.6, 0}], Text["\", {0, 0, 2.1}], {AbsoluteThickness[ .75], RGBColor[0, 0.5, 0], Line[{{\(-2\), 0, 0}, {1.5, 0, 0}}]}, {AbsoluteThickness[2], RGBColor[0, 0.5, 0], Line[{{0, \(-1.5\), 0}, {0, 1.5, 0}}], {RGBColor[0, 0.5, 0], Line[{{0, 0, \(-70\)}, {0, 0, 2}}]}}}];\)\), "\n", \(\(Show[%%, %, Axes \[Rule] False, Ticks \[Rule] None, Boxed \[Rule] False, ViewPoint \[Rule] {20, 10, 12}, DefaultFont \[Rule] 12. , ImageSize \[Rule] 1200, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 5: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 5.1", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(co[x_, n_] := Table[Subscript[x, j], {j, n}]\), "\[IndentingNewLine]", \(\(xv = co[x, 2];\)\), "\[IndentingNewLine]", \(\(hv = co[h, 2];\)\), "\[IndentingNewLine]", \(Print[\*"\"\\""]\ \), "\[IndentingNewLine]", \(g[{x1_, x2_}] := 3 x1^2 + 24 x1 - x2^2 + 36\), "\[IndentingNewLine]", \(Simplify[{\[PartialD]\_\(x\_1\)g[xv], \[PartialD]\_\(x\_2\)\ g[xv]} . hv == {\[PartialD]\_\(x\_1\)g[xv], \[PartialD]\_\(x\_2\)\ g[xv]} . xv]\), "\[IndentingNewLine]", \(\((3 x\_1 + 12)\) h\_1 - \(x\_2\) h\_2 \[Equal] 3 x\_1^2 + 12 x\_1 - x\_2^2\), "\[IndentingNewLine]", \(\(\(Simplify[%, %%]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(phi[t_] := {\(-4\) + 2 Cosh[t], 2 Sqrt[3] Sinh[t]}\), "\[IndentingNewLine]", \(phi[t] + \[Lambda] \[PartialD]\_t\ phi[t]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.2", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Eliminate::ifun]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\[Phi][a_, b_, c_]\)[y1_, y2_] := {a\ Sinh[y1] Cos[y2], b\ Sinh[y1] Sin[y2], c\ Cosh[y1]}\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot3D[ Append[\(\[Phi][3, 2, 2]\)[y1, y2], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {y1, 0, 2}, {y2, \(-\[Pi]\), \[Pi]}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {81, 81}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot3D[ Append[\(\[Phi][3, 2, \(-2\)]\)[y1, y2], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {y1, 0, 2}, {y2, \(-\[Pi]\), \[Pi]}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {81, 81}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(\(Show[p1, p2, PlotRange \[Rule] All, Axes \[Rule] False, Boxed \[Rule] False, ViewPoint \[Rule] {2, 1, 0.7}, Background \[Rule] lightblue, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\ \), "\ \[IndentingNewLine]", \(FullSimplify[ Eliminate[\(\[Phi][a, b, c]\)[y1, y2] \[Equal] {x\_1, x\_2, x\_3}, {y1, y2}]]\), "\[IndentingNewLine]", \(g[x1_, x2_, x3_] := \(-x1^2\)/a^2 - x2^2/b^2 + x3^2/c^2 - 1\), "\[IndentingNewLine]", \(Simplify[ a^2 b^2 c^2 g[x\_1, x\_2, x\_3] \[Equal] 0]\), "\[IndentingNewLine]", \(g[x\_1, x\_2, x\_3]\), "\[IndentingNewLine]", \(Print[\*"\"\\""]\ \), "\[IndentingNewLine]", \(FullSimplify[{\[PartialD]\_\(x\_1\)\ g[x\_1, x\_2, x\_3], \[PartialD]\_\(x\_2\)\ g[x\_1, x\_2, x\_3], \[PartialD]\_\(x\_3\)\ g[x\_1, x\_2, x\_3]} . \(({h\_1, h\_2, h\_3} - {x\_1, x\_2, x\_3})\) \[Equal] 0]\), "\[IndentingNewLine]", \(Print["\"]\ \), "\ \[IndentingNewLine]", \(\(\[Phi][a, b, c]\)[y\_1, y\_2]\), "\[IndentingNewLine]", \(Print["\"]\ \), "\ \[IndentingNewLine]", \({h1, h2, h3} = \(\[Phi][a, b, c]\)[y\_1, y\_2] + \[Lambda] \[PartialD]\_\(y\_1\)\(\[Phi][a, b, c]\)[y\_1, y\_2] + \[Mu] \[PartialD]\_\(y\_2\)\(\[Phi][a, b, c]\)[y\_1, y\_2]\), "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(\(sol = Solve[g[x1, x2, x3] \[Equal] 0, x3];\)\), "\[IndentingNewLine]", \(\(f1[x1_, x2_] = x3 /. sol[\([2]\)];\)\), "\[IndentingNewLine]", \(\(f2[x1_, x2_] = x3 /. sol[\([1]\)];\)\), "\[IndentingNewLine]", \(f1[x\_1, x\_2]\), "\[IndentingNewLine]", \(f2[x\_1, x\_2]\), "\[IndentingNewLine]", \(Print["\"]\ \), "\ \[IndentingNewLine]", \(FullSimplify[{x\_1, x\_2, f1[x\_1, x\_2]} + \[Lambda] {1, 0, \[PartialD]\_\(x\_1\)f1[x\_1, x\_2]} + \[Mu] {0, 1, \[PartialD]\_\(x\_2\)f1[x\_1, x\_2]}]\), "\[IndentingNewLine]", \(FullSimplify[{x\_1, x\_2, f2[x\_1, x\_2]} + \[Lambda] {1, 0, \[PartialD]\_\(x\_1\)f2[x\_1, x\_2]} + \[Mu] {0, 1, \[PartialD]\_\(x\_2\)f2[x\_1, x\_2]}]\), "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(FullSimplify[{\[PartialD]\_\(x\_1\)\ g[x\_1, x\_2, x\_3], \[PartialD]\_\(x\_2\)\ g[x\_1, x\_2, x\_3], \[PartialD]\_\(x\_3\)\ g[x\_1, x\_2, x\_3]} . \(({h1, h2, h3} - {x\_1, x\_2, x\_3})\) /. Thread[{x\_1, x\_2, x\_3} -> \(\[Phi][a, b, c]\)[y\_1, y\_2]]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.3", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(Phi[{x1_, x2_}] := {a/b\ x2, \(-b\)/a\ x1}\), "\[IndentingNewLine]", \(Nest[Phi, {x\_1, x\_2}, 4]\), "\[IndentingNewLine]", \(\(t = 3\ \[Pi]/4;\)\), "\n", \(\(a = 2;\)\), "\n", \(\(b = 1;\)\), "\n", \(\(x1 = a\ Cos[t];\)\), "\n", \(\(x2 = b\ Sin[t];\)\), "\n", \(\(p = {x1, x2};\)\), "\n", \(\(q = {a/b\ x2, \(-b\)/a\ x1};\)\), "\n", \(\(p1 = Graphics[{Text["\", p], Text["\", \(-p\)], Text["\", q], Text["\", \(-q\)], {AbsolutePointSize[5], darkblue, Point[p], Point[\(-p\)], Point[q], Point[\(-q\)]}, {RGBColor[1, 0, 0], Circle[{0, 0}, {a, b}]}}, AspectRatio \[Rule] Automatic];\)\), "\n", \(\(p2 = Graphics[{darkblue, {Line[{{x1 - a\ Sin[t], x2 + b\ Cos[t]}, {x1 + a\ Sin[t], x2 - b\ Cos[t]}}]}, Line[{{\(-x1\) - a\ Sin[t], \(-x2\) + b\ Cos[t]}, {\(-x1\) + a\ Sin[t], \(-x2\) - b\ Cos[t]}}], Line[{{a/b\ x2 - a\ Sin[t + \[Pi]/2], \(-b\)/a\ x1 + b\ Cos[t + \[Pi]/2]}, {a/b\ x2 + a\ Sin[t + \[Pi]/2], \(-b\)/a\ x1 - b\ Cos[t + \[Pi]/2]}}], Line[{{\(-a\)/b\ x2 - a\ Sin[t + \[Pi]/2], b/a\ x1 + b\ Cos[t + \[Pi]/2]}, {\(-a\)/b\ x2 + a\ Sin[t + \[Pi]/2], b/a\ x1 - b\ Cos[t + \[Pi]/2]}}]}];\)\), "\n", \(\(Show[p1, p2, DefaultFont \[Rule] 12. , Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.6", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex5.6.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.11", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Missing. PV409", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.18 (Lemniscate) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(c = {1/Sqrt[2], 0};\)\), "\[IndentingNewLine]", \(norm[x_] := Sqrt[x[\([1]\)]^2 + x[\([2]\)]^2]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Expand[ norm[\(({x\_1, x\_2} - c)\)]^2 norm[\(({x\_1, x\_2} + c)\)]^2 - norm[c]^4]\), "\[IndentingNewLine]", \(g[x1_, x2_] := norm[{x1, x2}]^4 - x1^2 + x2^2\), "\[IndentingNewLine]", \(Simplify[%% \[Equal] g[x\_1, x\_2]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Flatten[ Solve[{{\[PartialD]\_\(x\_1\)\ g[x\_1, x\_2], \[PartialD]\_\(x\_2\)\ g[x\_1, x\_2]} \[Equal] {0, 0}, g[x\_1, x\_2] \[Equal] 0}, {x\_1, x\_2}]] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ g[x1, x2] \[Equal] 0 /. Thread[{x1, x2} \[Rule] r {\ Cos[\[Alpha]], \ Sin[\[Alpha]]}], r > 0]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(sol[r_] = FullSimplify[ Solve[{g[x\_1, x\_2] \[Equal] 0, x\_1^2 - x\_2^2 == r^4}, {x\_1, x\_2}], r > 0];\)\), "\[IndentingNewLine]", \(Part[sol[r], {3, 4}] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]", \(\(\[Phi]\_+\)[ r_] := {x\_1, x\_2} /. \(sol[r]\)[\([4]\)]\), "\[IndentingNewLine]", \(\(\[Phi]\_+\)[r]\), "\[IndentingNewLine]", \(\(\[Phi]\_-\)[ r_] := {x\_1, x\_2} /. \(sol[r]\)[\([3]\)]\), "\[IndentingNewLine]", \(\(\[Phi]\_-\)[r]\), "\[IndentingNewLine]", \(ArcCos[\[PartialD]\_r\ \(\[Phi]\_+\)[r] . \[PartialD]\_r\ \(\[Phi]\_-\)[ r]/\((norm[\[PartialD]\_r\ \(\[Phi]\_+\)[r]] norm[\[PartialD]\_r\ \(\[Phi]\_-\)[r]])\) /. r \[Rule] 0]\), "\[IndentingNewLine]", \(\(ParametricPlot[ Evaluate[{\(\[Phi]\_+\)[r], \(\[Phi]\_-\)[r]}], {r, \(-1\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotPoints \[Rule] 200, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, ImageSize \[Rule] 800];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercises 5.19 (Astroid) and 5.20 (Cissoid) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[{Cos[t]^3, Sin[t]^3}, {t, \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotPoints \[Rule] 200, AspectRatio \[Rule] Automatic, Ticks \[Rule] None, PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[{{1 - y, Sqrt[\((\((1 - y)\)^3)\)/\((1 + y)\)]}, {1 - y, \(-Sqrt[\((\((1 - y)\)^3)\)/\((1 + y)\)]\)}}, {y, \(-0.8\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotPoints \[Rule] 200, PlotRange \[Rule] {{\(-1\), 3}, {\(-5.5\), 5.5}}, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = ParametricPlot[{{1 - y, Sqrt[\((\((1 - y)\)^3)\)/\((1 + y)\)]}, {1 - y, \(-Sqrt[\((\((1 - y)\)^3)\)/\((1 + y)\)]\)}}, {y, \(-0.2385\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], PlotPoints \[Rule] 100, PlotRange \[Rule] {{\(-1\), 3}, Automatic}, AspectRatio \[Rule] Automatic, Ticks \[Rule] {{0, 1}, {1}}, PlotStyle \[Rule] AbsoluteThickness[1.5], DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p4 = Graphics[{{AbsoluteThickness[2], RGBColor[1, 0, 0], Circle[{1, 0}, 1, {\(-0.5\)\ \[Pi], 0.5\ \[Pi]}]}, {Dashing[{0.025, 0.025}], RGBColor[1, 0, 0], Circle[{1, 0}, 1]}, {Dashing[{0.045, 0.045}], darkblue, Line[{{2, 5.5}, {2, \(-5.5\)}}]}}];\)\), "\n", \(\(p5 = Show[p2, p3, p4, PlotRange \[Rule] {{\(-1\), 2.5}, {\(-3\), 3}}, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[GraphicsArray[{p1, p5}], Background \[Rule] lightblue, ImageSize \[Rule] 1000, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.21 (Conchoid and trisection of angle) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Plot::plnr]\), "\n", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(d = 0.4;\)\), "\n", \(\(p1 = Graphics[{Text["\<0\>", {\(-0.1\), \(-0.6\)}], Text["\", {1.2, 1}], {Dashing[{0.025, 0.025}], darkblue, Line[{{1, \(-10\)}, {1, 10}}]}, {Dashing[{0.02, 0.02}], darkblue, Line[{{0, 0}, {1.5, 2.25}}]}, {AbsolutePointSize[4], darkblue, Point[{1, 1.5}], Point[{0.77812, 1.16718}], Point[{1.22188, 1.83282}]}}];\)\), "\n", \(\(p2 = Plot[{Sqrt[\((\(-x^4\) + 2\ x^3 + \((d^2 - 1)\)\ x^2)\)/\((x^2 - 2\ x + 1)\)], \(-Sqrt[\((\(-x^4\) + 2\ x^3 + \((d\ d - 1)\)\ x^2)\)/\((x^2 - 2\ x + 1)\)]\)}, {x, d, 1 + d}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], AxesOrigin \[Rule] {0, 0}, Axes \[Rule] True, Ticks \[Rule] None, PlotPoints \[Rule] 50, PlotRange \[Rule] {{\(-0.25\), 2}, {\(-5\), 5}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = Show[p2, p1, AspectRatio \[Rule] 1, DefaultFont \[Rule] 12. , DisplayFunction \[Rule] Identity];\)\), "\n", \(\(conchoid[d_]\)[ s_] := \((\(-d\) + Cosh[s])\)/\((Cosh[s])\) {1, Sinh[s]}\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot[ Evaluate[ Table[\(conchoid[d]\)[s], {d, 1/2, 3/2, 1/10}]], {s, \(-1.5\), 1.5}, PlotStyle \[Rule] RGBColor[1, 0, 0], AspectRatio \[Rule] Automatic, Axes \[Rule] None, PlotPoints \[Rule] 100, PlotRange \[Rule] {{\(-0.6\), 0.6}, Automatic}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[GraphicsArray[{p3, p4}], Background \[Rule] lightblue, ImageSize \[Rule] 1000, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.22 (Villarceau's circles) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[{Cos[t]\ \((2 + Cos[u])\), Sin[t]\ \((2 + Cos[u])\), Sin[u]}, {t, \(-\[Pi]\), \[Pi]}, {u, \(-\[Pi]\), \[Pi]}, PlotPoints \[Rule] 50, Axes \[Rule] False, Boxed \[Rule] False, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{{AbsoluteThickness[1], darkblue, Circle[{2, 0}, 1], Circle[{\(-2\), 0}, 1], Circle[{0, \(-5\)}, 3], Circle[{0, \(-5\)}, 1]}, {darkblue, AbsoluteThickness[0.5], Line[{{\(-2\), 1}, {2, 1}}], Line[{{\(-2\), \(-1\)}, {2, \(-1\)}}]}, {darkblue, AbsolutePointSize[5], Point[{1.5, 0.5\ Sqrt[3]}], Point[{\(-1.5\), \(-0.5\)\ Sqrt[3]}], Point[{1.5, \(-5\)}]}, {AbsolutePointSize[5], GrayLevel[0.25], Point[{\(-1.5\), \(-5\)}]}, {darkblue, AbsoluteThickness[0.5], Line[{{\(-3.5\), \(-3.5\)/Sqrt[3]}, {3, 3/Sqrt[3]}}]}, {darkblue, AbsoluteThickness[1.5], Circle[{0, \(-6\)}, {Sqrt[3], 2}, {\(-0.5\)\ \[Pi], 0.5\ \[Pi]}], Circle[{0, \(-4\)}, {Sqrt[3], 2}, {\(-0.5\)\ \[Pi], 0.5\ \[Pi]}]}, {darkblue, AbsoluteThickness[1.5], Dashing[{0.02, 0.02}], Circle[{0, \(-6\)}, {Sqrt[3], 2}, {0.5\ \[Pi], 1.5\ \[Pi]}], Circle[{0, \(-4\)}, {Sqrt[3], 2}, {0.5\ \[Pi], 1.5\ \[Pi]}]}, {RGBColor[1, 0, 0], AbsoluteThickness[0.5], Dashing[{0.015, 0.015}], Line[{{3, 0}, {3, \(-5\)}}], Line[{{\(-3\), 0}, {\(-3\), \(-5\)}}], Line[{{1, 0}, {1, \(-5\)}}], Line[{{\(-1\), 0}, {\(-1\), \(-5\)}}], Line[{{1.5, 0.5\ Sqrt[3]}, {1.5, \(-5\)}}], Line[{{\(-1.5\), \(-0.5\)\ Sqrt[3]}, {\(-1.5\), \(-5\)}}]}, Text["\", {\(-3\), 1}], Text["\", {3, 2}], Text["\<-q\>", {\(-2\), \(-5\)}], Text["\", {2, \(-5\)}]}, {AspectRatio \[Rule] Automatic, DefaultFont \[Rule] 12. }];\)\), "\n", \(\(Show[GraphicsArray[{%%, %}], Background \[Rule] lightblue, ImageSize \[Rule] 1000, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.23 (Torus sections) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Missing. PV426N", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Exercise 5.26 (Cauchy-Schwarz inequality in ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "n"], TraditionalForm]]], ", Grassmann's, Jacobi's and Lagrange's identities in ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "3"], TraditionalForm]]], ") " }], "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(vec[v_] := Table[v\_j, {j, 3}]\), "\[IndentingNewLine]", \(\(v1 = vec[v\_1];\)\), "\[IndentingNewLine]", \(\(v2 = vec[v\_2];\)\), "\[IndentingNewLine]", \(\(v3 = vec[v\_3];\)\), "\[IndentingNewLine]", \(\(v4 = vec[v\_4];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Simplify[Cross[v1, Cross[v2, v3]]]\), "\[IndentingNewLine]", \(Simplify[ Cross[v1, Cross[v2, v3]] \[Equal] v3 . v1\ v2 - v1 . v2\ v3]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Cross[v1, Cross[v2, v3]] + Cross[v2, Cross[v3, v1]] + Cross[v3, Cross[v1, v2]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Cross[Cross[v1, v2], Cross[v2, v3]]\), "\[IndentingNewLine]", \(Det[{v4, v1, v2}] v3 - Det[{v1, v2, v3}] v4\), "\[IndentingNewLine]", \(Simplify[ Cross[Cross[v1, v2], Cross[v3, v4]] == Det[{v4, v1, v2}] v3 - Det[{v1, v2, v3}] v4]\ \[IndentingNewLine]\), "\[IndentingNewLine]", \(Cross[v1, v2] . Cross[v3, v4]\), "\[IndentingNewLine]", \(Det[{{v1 . v3, v1 . v4}, {v2 . v3, v2 . v4}}]\), "\[IndentingNewLine]", \(Simplify[ Cross[v1, v2] . Cross[v3, v4] \[Equal] Det[{{v1 . v3, v1 . v4}, {v2 . v3, v2 . v4}}]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.27 (Spherical trigonometry)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["\<\ The code below might be improved. Formulae have been added by means \ of Xfig.\ \>", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(\(Clear["\<`*\>"];\)\), "\[IndentingNewLine]", \(Off[General::spell]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\[IndentingNewLine]\[IndentingNewLine] (*\ transparant\ sphere\ *) \), "\[IndentingNewLine]", \(\(x1[a_, t_] := Cos[a]*Cos[t];\)\), "\n", \(\(x2[a_, t_] := Sin[a]*Cos[t];\)\), "\n", \(\(x3[a_, t_] := Sin[t];\)\), "\n", \(\(\(x[a_, t_] := {x1[a, t], x2[a, t], x3[a, t]};\)\(\n\) \)\), "\[IndentingNewLine]", \(\(raster = 0.075;\)\), "\n", \(\(point = 0.001;\)\), "\n", \(\(list = Table[Point[x[a, t]], {a, \(-\[Pi]\), \[Pi], raster}, {t, \(-\[Pi]\)/2, \[Pi]/2, raster}];\)\), "\n", \(\(S = Graphics3D[{PointSize[point], list}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False];\)\[IndentingNewLine]\n (*\ three\ points\ axis\ A, B, C\ on\ unit\ sphere\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(A = x[\[Pi]/3, \[Pi]/3];\)\), "\n", \(\(A1 = A[\([1]\)];\)\), "\n", \(\(A2 = A[\([2]\)];\)\), "\n", \(\(\(A3 = A[\([3]\)];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(B = x[\(-\[Pi]\)/10, \[Pi]/10];\)\), "\n", \(\(B1 = B[\([1]\)];\)\), "\n", \(\(B2 = B[\([2]\)];\)\), "\n", \(\(\(B3 = B[\([3]\)];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(CC = x[\[Pi]/3, \(-\[Pi]\)/7];\)\), "\n", \(\(CC1 = CC[\([1]\)];\)\), "\n", \(\(CC2 = CC[\([2]\)];\)\), "\n", \(\(\(CC3 = CC[\([3]\)];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(norm[a1_, a2_, a3_] := N[Sqrt[\((a1)\)^2 + \((a2)\)^2 + \((a3)\)^2]];\)\[IndentingNewLine]\n \ (*\ matrix\ of\ rotation\ about\ axis\ {a1, a2, a3}\ by\ angle\ a\ *) \), "\[IndentingNewLine]", \(\(R[a_, a1_, a2_, a3_] := {{Cos[a] + a1^2 \((1 - Cos[a])\), \(-a3\)*Sin[a] + a1*a2*\((1 - Cos[a])\), a2*Sin[a] + a1*a3*\((1 - Cos[a])\)}, {a3*Sin[a] + a1*a2*\((1 - Cos[a])\), Cos[a] + a2^2 \((1 - Cos[a])\), \(-a1\)*Sin[a] + a2*a3*\((1 - Cos[a])\)}, {\(-a2\)*Sin[a] + a1*a3*\((1 - Cos[a])\), a1*Sin[a] + a2*a3*\((1 - Cos[a])\), Cos[a] + a3^2 \((1 - Cos[a])\)}};\)\n\[IndentingNewLine] (*\ vector\ r\ on\ unit\ sphere\ perpendicular\ to\ p\ and\ q, \ that\ is, \ normalized\ cross\ product\ of\ p\ and\ q\ *) \), \ "\[IndentingNewLine]", \(\(Crossproduct[p1_, p2_, p3_, q1_, q2_, q3_] := {p1, p2, p3}\[Cross]{q1, q2, q3};\)\), "\n", \(\(rr1[p1_, p2_, p3_, q1_, q2_, q3_] := \(Crossproduct[p1, p2, p3, q1, q2, q3]\)[\([1]\)];\)\), "\n", \(\(rr2[p1_, p2_, p3_, q1_, q2_, q3_] := \(Crossproduct[p1, p2, p3, q1, q2, q3]\)[\([2]\)];\)\), "\n", \(\(rr3[p1_, p2_, p3_, q1_, q2_, q3_] := \(Crossproduct[p1, p2, p3, q1, q2, q3]\)[\([3]\)];\)\), "\n", \(\(Nnorm[p1_, p2_, p3_, q1_, q2_, q3_] := norm[rr1[p1, p2, p3, q1, q2, q3], rr2[p1, p2, p3, q1, q2, q3], rr3[p1, p2, p3, q1, q2, q3]];\)\), "\n", \(\(r1[p1_, p2_, p3_, q1_, q2_, q3_] := rr1[p1, p2, p3, q1, q2, q3]/ Nnorm[p1, p2, p3, q1, q2, q3];\)\), "\n", \(\(r2[p1_, p2_, p3_, q1_, q2_, q3_] := rr2[p1, p2, p3, q1, q2, q3]/ Nnorm[p1, p2, p3, q1, q2, q3];\)\), "\n", \(\(r3[p1_, p2_, p3_, q1_, q2_, q3_] := rr3[p1, p2, p3, q1, q2, q3]/ Nnorm[p1, p2, p3, q1, q2, q3];\)\[IndentingNewLine]\[IndentingNewLine] (*\ as\ a\ control, \ norm\ of\ perpendicular\ vector\ R\ and\ inproduct\ of\ R\ and\ A\ *) \ \), "\n", \(\(N[ norm[r1[A1, A2, A3, B1, B2, B3], r2[A1, A2, A3, B1, B2, B3], r3[A1, A2, A3, B1, B2, B3]]];\)\), "\n", \(\(N[{A1, A2, A3} . {r1[A1, A2, A3, B1, B2, B3], r2[A1, A2, A3, B1, B2, B3], r3[A1, A2, A3, B1, B2, B3]}];\)\[IndentingNewLine]\[IndentingNewLine] (*\ geodesic\ connecting\ p\ and\ q\ arising\ as\ image\ under\ rotation\ \ about\ r\ *) \), "\n", \(\(geod[s_, p1_, p2_, p3_, q1_, q2_, q3_] := R[s, r1[p1, p2, p3, q1, q2, q3], r2[p1, p2, p3, q1, q2, q3], r3[p1, p2, p3, q1, q2, q3]] . {p1, p2, p3};\)\), "\n", \(\(angle[p1_, p2_, p3_, q1_, q2_, q3_] := ArcCos[{p1, p2, p3} . {q1, q2, q3}];\)\n\[IndentingNewLine] (*\ as\ a\ check, the\ geodesic\ must\ start\ at\ p\ and\ end\ at\ q\ *) \), "\ \[IndentingNewLine]", \(\(N[geod[0, A1, A2, A3, B1, B2, B3]];\)\), "\n", \(\(N[ geod[angle[A1, A2, A3, B1, B2, B3], A1, A2, A3, B1, B2, B3]];\)\[IndentingNewLine]\n (*\ graphics\ of\ the\ geodesic\ *) \), "\[IndentingNewLine]", \(\(\(GEOD[p1_, p2_, p3_, q1_, q2_, q3_] := ParametricPlot3D[ Evaluate[geod[s, p1, p2, p3, q1, q2, q3]], {s, 0, angle[p1, p2, p3, q1, q2, q3]}, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(GEOAB = GEOD[A1, A2, A3, B1, B2, B3];\)\), "\n", \(\(GEOBC = GEOD[B1, B2, B3, CC1, CC2, CC3];\)\), "\n", \(\(\(GEOCA = GEOD[CC1, CC2, CC3, A1, A2, A3];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(dash = 0.007;\)\), "\n", \(\(thick = 0.002;\)\), "\n", \(\(LA = Graphics3D[{Thickness[thick], Dashing[{dash, 2 dash}], Line[{{0, 0, 0}, {A1, A2, A3}}]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(LB = Graphics3D[{Thickness[thick], Dashing[{dash, 2 dash}], Line[{{0, 0, 0}, {B1, B2, B3}}]}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(LC = Graphics3D[{Thickness[thick], Dashing[{dash, 2 dash}], Line[{{0, 0, 0}, {CC1, CC2, CC3}}]}, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine]\n (*\ unit\ tangent\ vector\ of\ geodesic\ connecting\ p\ and\ q\ at\ p\ *) \ \), "\[IndentingNewLine]", \(\(\(tang[s_, p1_, p2_, p3_, q1_, q2_, q3_] := \((s/ Nnorm[p1, p2, p3, q1, q2, q3])\)*\(({q1, q2, q3} - {p1, p2, p3} . {q1, q2, q3}\ {p1, p2, p3})\);\)\(\n\) \)\), "\[IndentingNewLine]", \(\(\(TA[p1_, p2_, p3_, q1_, q2_, q3_] := ParametricPlot3D[ Evaluate[{p1, p2, p3} + tang[s, p1, p2, p3, q1, q2, q3]], {s, 0, 0.6}, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(TAAB = TA[A1, A2, A3, B1, B2, B3];\)\), "\n", \(\(TABA = TA[B1, B2, B3, A1, A2, A3];\)\), "\n", \(\(TABC = TA[B1, B2, B3, CC1, CC2, CC3];\)\), "\n", \(\(TACB = TA[CC1, CC2, CC3, B1, B2, B3];\)\), "\n", \(\(TACA = TA[CC1, CC2, CC3, A1, A2, A3];\)\), "\n", \(\(TAAC = TA[A1, A2, A3, CC1, CC2, CC3];\)\[IndentingNewLine]\n (*\ angles\ between\ tangent\ lines\ *) \[IndentingNewLine] (*\ \(N[ ArcCos[tang[1, A1, A2, A3, B1, B2, B3] . tang[1, A1, A2, A3, CC1, CC2, CC3]]/\[Pi]];\)\ *) \[IndentingNewLine] (*\ \(N[ ArcCos[tang[1, B1, B2, B3, A1, A2, A3] . tang[1, B1, B2, B3, CC1, CC2, CC3]]/\[Pi]];\)\ *) \[IndentingNewLine] (*\ \(N[ ArcCos[tang[1, CC1, CC2, CC3, A1, A2, A3] . tang[1, CC1, CC2, CC3, B1, B2, B3]]/\[Pi]];\)\ *) \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(ANGA = ParametricPlot3D[ Evaluate[{A1, A2, A3} + \((Cos[a]/2)\)* tang[1, A1, A2, A3, B1, B2, B3] + \((Sin[a]/2)\)* tang[1, A1, A2, A3, CC1, CC2, CC3]], {a, 0, \[Pi]/2}, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(ANGB = ParametricPlot3D[ Evaluate[{B1, B2, B3} + \((Cos[a]/2)\)* tang[1, B1, B2, B3, A1, A2, A3] + \((Sin[a]/2)\)* tang[1, B1, B2, B3, CC1, CC2, CC3]], {a, 0, \[Pi]/2}, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(ANGC = ParametricPlot3D[ Evaluate[{CC1, CC2, CC3} + \((Cos[a]/2)\)* tang[1, CC1, CC2, CC3, A1, A2, A3] + \((Sin[a]/2)\)* tang[1, CC1, CC2, CC3, B1, B2, B3]], {a, 0, 1.558}, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\n\[IndentingNewLine] (*\ final\ \(result : plot\ of\ three\ geodesics\ and\ six\ tangent\ lines\ on\ \ transparent\ sphere\ and\ the\ angles\ between\ the\ tangent\ lines\)\ *) \), \ "\[IndentingNewLine]", \(\(f = 10;\)\), "\n", \(\(view = f {A1 + B1 + CC1, A2 + B2 + CC2, A3 + B3 + CC3 + 1/8}/ norm[A1 + B1 + CC1, A2 + B2 + CC2, A3 + B3 + CC3 + 1/8];\)\), "\n", \(\(Show[GEOAB, GEOBC, GEOCA, S, LA, LB, LC, TAAB, TABA, TABC, TACB, TACA, TAAC, ANGA, ANGB, ANGC, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] view, ImageSize \[Rule] 900, DefaultColor \[Rule] darkblue, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\), "\n", \(\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Angle between two planes equals angle between normals ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(<< LinearAlgebra`Orthogonalization`\), "\[IndentingNewLine]", \(\(a1 = {1, 0, 0};\)\), "\[IndentingNewLine]", \(\(a2 = {0, 1, 0};\)\), "\[IndentingNewLine]", \(\(a3 = \(1\/\@3\) {1, \(-1\), \(-1\)};\)\), "\[IndentingNewLine]", \(\(a4 = Normalize[a1 - \(\@3\) a3];\)\), "\[IndentingNewLine]", \(\(a5 = Normalize[Cross[a1, a2]];\)\), "\[IndentingNewLine]", \(\(a6 = Normalize[Cross[a1, a4]];\)\), "\[IndentingNewLine]", \(a4 . a1\), "\[IndentingNewLine]", \(a4 . a2\), "\[IndentingNewLine]", \(a5 . a6\), "\[IndentingNewLine]", \(\(thick = 0.003;\)\), "\[IndentingNewLine]", \(\(l1 = Graphics3D[{Thickness[thick], RGBColor[1, 0, 0], Line[{{0, 0, 0}, a2}]}];\)\), "\[IndentingNewLine]", \(\(l2 = Graphics3D[{Thickness[thick], RGBColor[1, 0, 0], Line[{{0, 0, 0}, a4}]}];\)\), "\[IndentingNewLine]", \(\(l3 = Graphics3D[{Thickness[thick], RGBColor[1, 0, 0], Line[{{0, 0, 0}, a5}]}];\)\), "\[IndentingNewLine]", \(\(l4 = Graphics3D[{Thickness[thick], RGBColor[1, 0, 0], Line[{{0, 0, 0}, a6}]}];\)\), "\[IndentingNewLine]", \(\(pl1 = ParametricPlot3D[ Append[\[Lambda]\ a1 + \[Mu]\ a2, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Lambda], \(-1\), 1}, {\[Mu], \(-0.3\), 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] 40, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(pl2 = ParametricPlot3D[ Append[\[Lambda]\ a1 + \[Mu]\ a4, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Lambda], 0, 1}, {\[Mu], \(-0.3\), 1}, PlotPoints \[Rule] 40, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[l1, l2, l3, l4, pl1, pl2, ViewPoint \[Rule] 5 {\(-2\), \(-1\), 0.5}, Boxed \[Rule] False, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.29 (Stereographic projection)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(\(Clear["\<`*\>"];\)\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(x1[a_, t_] := Cos[a] Cos[t];\)\), "\n", \(\(x2[a_, t_] := Sin[a] Cos[t];\)\), "\n", \(\(x3[a_, t_] := Sin[t];\)\), "\n", \(\(\(x[a_, t_] := {x1[a, t], x2[a, t], 1 + x3[a, t]};\)\(\n\) \)\), "\[IndentingNewLine]", \(\(y1[a_, t_] := 2 Cos[a] Cos[t]/\((1 - Sin[t])\);\)\), "\n", \(\(y2[a_, t_] := 2 Sin[a] Cos[t]/\((1 - Sin[t])\);\)\), "\n", \(\(y[a_, t_] := {y1[a, t], y2[a, t], 0};\)\n\[IndentingNewLine] (*\ \[IndentingNewLine]p1 = ParametricPlot3D[ Evaluate[ x[a, t]], {a, \(-\[Pi]\), \[Pi]}, {t, \(-\[Pi]\)/2 + 0.01, \[Pi]/2 - 0.01}, PlotPoints \[Rule] 30, DisplayFunction \[Rule] Identity]; \n p2 = ParametricPlot3D[ Evaluate[ y[a, t]], {a, \(-\[Pi]\), \[Pi]}, {t, \(-\[Pi]\)/2 + 0.01, \[Pi]/2 - 0.01}, PlotPoints \[Rule] 30, DisplayFunction \[Rule] Identity]; \n l1 = Graphics3D[{Thickness[0.003], Line[{x[\[Pi]/4, \[Pi]/3], y[\[Pi]/4, \[Pi]/3]}]}, Shading \[Rule] False]; \n l2 = Graphics3D[{Thickness[0.003], Line[{x[\(-\[Pi]\)/6, \[Pi]/6], y[\(-\[Pi]\)/6, \[Pi]/6]}]}, Shading \[Rule] False]; \ \n Show[p1, p2, l1, l2, ViewPoint \[Rule] {3, 1, 0.5}, Axes \[Rule] None, Ticks \[Rule] False, Boxed \[Rule] False, Shading \[Rule] False, DisplayFunction \[Rule] $DisplayFunction];\ \[IndentingNewLine]*) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(raster = 0.075;\)\), "\n", \(\(point = 0.001;\)\), "\n", \(\(list1 = Table[Point[x[a, t]], {a, \(-\[Pi]\), \[Pi], raster}, {t, \(-\[Pi]\)/2 + 0.01, \[Pi]/2 - 0.01, raster}];\)\), "\n", \(\(\(sphere = Graphics3D[{PointSize[point], list1}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(para = 0.9;\)\), "\n", \(\(circ[b_] := {para, Sqrt[1 - para^2]*Cos[b], 1 + Sqrt[1 - para^2]*Sin[b]};\)\), "\n", \(\(\(circ1 = ParametricPlot3D[Evaluate[circ[b]], {b, \(-\[Pi]\), \[Pi]}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(proj[b_] := {2*para/\((1 - Sqrt[1 - para^2]*Sin[b])\), 2*Sqrt[1 - para^2]*Cos[b]/\((1 - Sqrt[1 - para^2]*Sin[b])\), 0};\)\), "\n", \(\(\(circ2 = ParametricPlot3D[Evaluate[proj[b]], {b, \(-\[Pi]\), \[Pi]}, Axes \[Rule] False, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(dash = 0.007;\)\), "\n", \(\(l1 = Graphics3D[{RGBColor[1, 0, 0], Thickness[0.001], Dashing[{dash, dash}], Line[{{0, 0, 2}, proj[\[Pi]/2]}], Line[{{0, 0, 2}, proj[0.9 \[Pi]]}], Line[{{0, 0, 2}, proj[\[Pi]/5]}], Line[{{0, 0, 2}, proj[\(-\[Pi]\)/2]}]}];\)\), "\n", \(\(l2 = Graphics3D[{Thickness[0.001], Line[{{0, 0, 0}, 1.1 proj[\[Pi]/2]}]}];\)\), "\n", \(\(point = 0.007;\)\), "\n", \(\(p = Graphics3D[{PointSize[point], Point[circ[\[Pi]/2]], Point[circ[0.9 \[Pi]]], Point[circ[\[Pi]/5]], Point[circ[\(-\[Pi]\)/2]], Point[proj[\[Pi]/2]], Point[proj[0.9 \[Pi]]], Point[proj[\[Pi]/5]], Point[proj[\(-\[Pi]\)/2]]}];\)\), "\n", \(\(\(mer = ParametricPlot3D[Evaluate[x[0, t]], {t, \(-\[Pi]\)/2, \[Pi]/2}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(Show[circ1, circ2, sphere, mer, l1, l2, p, ViewPoint \[Rule] {3, 1.5, 1.5}, ImageSize \[Rule] 900, Background \[Rule] lightblue, DefaultColor \[Rule] darkblue, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.31 (Mercator projection of sphere onto cylinder)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(<< Miscellaneous`WorldPlot`;\)\), "\n", \(\(worldmap = WorldPlot[{World, RandomColors}, WorldProjection \[Rule] Mercator, WorldPoints \[Rule] 2000, WorldRange \[Rule] {{\(-60\), 82.5}, {\(-180\), 105}}, WorldBackground \[Rule] GrayLevel[0.96], WorldGrid \[Rule] {7.5, 7.5}, DisplayFunction \[Rule] Identity];\)\n\[IndentingNewLine] (*\ geographical\ coordinates\ Los\ Angeles\ \(33\^\[EmptySmallCircle]\) \ \(56\^'\)\ North\ and\ \(118\^\[EmptySmallCircle]\) \(24\^'\)\ West\ and\ \ Utrecht\ \(52\^\[EmptySmallCircle]\) \(06\^'\)\ North\ and\ \(02\^\ \[EmptySmallCircle]\) \(06\^'\)\ East\ *) \ \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\({LosA, Utre} = ToMinutes[{{{33, 56}, {\(-118\), \(-24\)}}, {{52, 06}, {05, 06}}}];\)\[IndentingNewLine]\n (*\ loxodrome\ is\ straight\ line\ on\ Mercator\ map\ *) \ \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(loxo = WorldGraphics[{RGBColor[1, 0, 0], Dashing[{0.005, 0.005}], Line[{LosA, Utre}]}];\)\n\n (*\ conversion\ from\ geographical\ coordinates\ to\ spherical\ coordinates\ \ in\ radians\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(<< Calculus`VectorAnalysis`\), "\[IndentingNewLine]", \(<< LinearAlgebra`Orthogonalization`\), "\[IndentingNewLine]", \(x[\[Alpha]_, \[Theta]_] := {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]]}\), "\[IndentingNewLine]", \(angle[d_, m_] := \((d + m\/60)\) \[Pi]\/180\), "\[IndentingNewLine]", \(\(\[Alpha]1 = \(-angle[118, 24]\);\)\), "\n", \(\(\[Theta]1 = angle[33, 56];\)\), "\n", \(\(\[Alpha]2 = angle[05, 06];\)\), "\n", \(\(\[Theta]2 = angle[52, 06];\)\), "\n", \(\(LA = N[x[\[Alpha]1, \[Theta]1]];\)\), "\n", \(\(UT = N[x[\[Alpha]2, \[Theta]2]];\)\[IndentingNewLine]\[IndentingNewLine] \ (*\ geodesic\ = \ great\ circle\ arises\ as\ orbit\ under\ rotation\ about\ axis\ \ perpendicular\ to\ LA\ and\ UT\ *) \ \[IndentingNewLine] (*\ distance\ and\ axis\ of\ rotatation\ *) \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(\[Delta] = N[ArcCos[LA . UT]];\)\), "\[IndentingNewLine]", \(\(n = \({a1, a2, a3} = N[Normalize[ Cross[LA, UT]]]\);\)\[IndentingNewLine]\[IndentingNewLine] (*\ matrix\ $R_ {\[Alpha]} $\ of\ rotation\ by\ angle\ $\[Alpha]$\ about\ \ axis\ $n$\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(r = {{0, \(-a3\), a2}, {a3, 0, \(-a1\)}, {\(-a2\), a1, 0}};\)\), "\[IndentingNewLine]", \(\(\(R[\[Alpha]_] := Cos[\[Alpha]] IdentityMatrix[3] + \((1 - Cos[\[Alpha]])\) Transpose[{n}] . {n} + Sin[\[Alpha]] r\)\(\[IndentingNewLine]\)\(\n\)\( (*\ point\ on\ geodesic\ issuing\ from\ point\ LA\ arising\ as\ image\ \ under\ rotation\ by\ angle\ $\[Alpha]$\ about\ axis\ $ \((a_ 1, a_ 2, a_ 3)\) $\ *) \)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(geod[\[Alpha]_] := R[\[Alpha]] . LA\)\(\n\)\(\[IndentingNewLine]\)\( (*\ the\ description\ of\ points\ in\ WorldPlot\ has\ to\ be\ in\ latitude\ \ and\ longitude\ given\ in\ degrees\ and\ minutes\ \ *) \)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(grminu[\[Theta]_] := 60 IntegerPart[\[Theta]] + IntegerPart[60 \((\[Theta] - IntegerPart[\[Theta]])\)];\)\), "\n", \(\(lat[\[Alpha]_] := grminu[\((180/\[Pi])\)\ ArcSin[\(geod[\[Alpha]]\)[\([3]\)]]];\)\), "\n\ ", \(\(lon[\[Alpha]_] := grminu[\((180/\[Pi])\)\ 2 ArcTan[\(geod[\[Alpha]]\)[\([2]\)]/\((\(geod[\[Alpha]]\)[\([1]\)]\ + Sqrt[\(geod[\[Alpha]]\)[\([1]\)]^2 + \ \(geod[\[Alpha]]\)[\([2]\)]^2])\)]];\)\), "\[IndentingNewLine]", \(\(Geodesic = Table[Point[{lat[\[Alpha]], lon[\[Alpha]]}], {\[Alpha], 0, \[Delta], 0.0001}];\)\), "\[IndentingNewLine]", \(\(\(geod = WorldGraphics[{PointSize[0.0006], Geodesic}];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(Show[{worldmap, loxo, geod}, ImageSize \[Rule] 800, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\)}], "Input"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(\[Theta] = \[Pi]/3.5;\)\), "\[IndentingNewLine]", \(p[\[Alpha]_] := {Cos[\[Alpha]], Sin[\[Alpha]]}\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[p[\[Alpha]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] darkblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[0.2 p[\[Alpha]], {\[Alpha], 0, \[Theta]}, PlotStyle -> RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(p3 = Graphics[{darkblue, PointSize[0.015], Point[0 p[0]], Point[p[\[Pi]/2]], Point[p[\[Theta]]], Point[{Tan[\[Theta]/2 + \[Pi]/4], 0}]}];\)\), "\[IndentingNewLine]", \(\(p4 = Graphics[{darkblue, Line[{p[\[Theta]], 0 p[0], p[\[Pi]/2], {Tan[\[Theta]/2 + \[Pi]/4], 0}, 0 p[0]}]}];\)\), "\[IndentingNewLine]", \(\(Show[p1, p2, p3, p4, Axes \[Rule] False, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.32 (Local isometry of catenoid and helicoid)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["Missing. P427V", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.34 (Hypo- and epicycloids) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(ParametricPlot[{3\ Cos[t] - Cos[3\ t], 3\ Sin[t] - Sin[3\ t]}, {t, 0, 2\ \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotPoints \[Rule] 50, AspectRatio \[Rule] Automatic, Ticks \[Rule] {{{2, {"\"}}}, {{4, {"\<2a\>"}}}}, Background \[Rule] lightblue, ImageSize \[Rule] 400];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.35 (Steiner's hypocycloid) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[ General::spell1]\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_] := b {2 Cos[\[Alpha]] + Cos[2 \[Alpha]], 2 Sin[\[Alpha]] - Sin[2 \[Alpha]]}\), "\[IndentingNewLine]", \(norm[{x1_, x2_}] := Sqrt[x1^2 + x2^2]\), "\[IndentingNewLine]", \(f[\[Alpha]_] := FullSimplify[norm[\[Phi][\[Alpha]]], b > 0]\), "\[IndentingNewLine]", \(f[\[Alpha]]\), \ "\[IndentingNewLine]", \(a = 3;\), "\[IndentingNewLine]", \(b = a/3;\), "\[IndentingNewLine]", \(Plot[ f[\[Alpha]], {\[Alpha], 0, 2 \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle -> RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] {Table[k \[Pi]\/3, {k, 0, 6}], {f[\[Pi]\/3], 2, f[0]}}, ImageSize \[Rule] 500];\), "\[IndentingNewLine]", \(Reduce[ f[\[Alpha]] \[Equal] a, \[Alpha]]\), "\[IndentingNewLine]", \(Reduce[ f[\[Alpha]] \[Equal] b, \[Alpha]]\), "\[IndentingNewLine]", \(Reduce[\[PartialD]\_\[Alpha]\ \ \[Phi][\[Alpha]] \[Equal] 0, \[Alpha]]\), "\[IndentingNewLine]", \(Simplify[ Sin[\[Alpha]] + Sin[2 \[Alpha]] == 2 Sin[3 \[Alpha]/2] Cos[\[Alpha]/2], Im[\[Alpha]] \[Equal] 0]\), "\[IndentingNewLine]", \(Simplify[ Cos[\[Alpha]] - Cos[2 \[Alpha]] \[Equal] 2 Sin[3 \[Alpha]/2] Sin[\[Alpha]/2], Im[\[Alpha]] \[Equal] 0]\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_\[Alpha]\ \ \[Phi][\[Alpha]] \[Equal] \(-4\) Sin[3 \[Alpha]/2] {Cos[\[Alpha]/ 2], \(-Sin[\[Alpha]/2]\)}, {Sin[\[Alpha]] + Sin[2 \[Alpha]] == 2 Sin[3 \[Alpha]/2] Cos[\[Alpha]/2], Cos[\[Alpha]] - Cos[2 \[Alpha]] \[Equal] 2 Sin[3 \[Alpha]/2] Sin[\[Alpha]/2], Im[\[Alpha]] \[Equal] 0}]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\) at \[Phi](\[Alpha])\>\"", "]"}], "\[IndentingNewLine]", \({Cos[\[Alpha]/ 2], \(-Sin[\[Alpha]/2]\)}\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\) at \[Phi](\[Alpha])\>\"", "]"}], "\[IndentingNewLine]", \(sol = Solve[{Cos[\[Alpha]/2], \(-Sin[\[Alpha]/2]\)} . {1, kk} \[Equal] 0, kk];\), "\[IndentingNewLine]", \(k = kk /. sol;\), "\[IndentingNewLine]", \({k1, k2} = Sin[\[Alpha]/2] Flatten[{1, k}]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\) in \[Phi](\[Alpha])\>\"", "]"}], "\[IndentingNewLine]", \(Clear[b]\), "\[IndentingNewLine]", RowBox[{\(Simplify[{h\_1, h\_2} . {k1, k2} \[Equal] \[Phi][\[Alpha]] . {k1, k2}]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Clear["\<`*\>"]\), "\ \[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\[IndentingNewLine]", \(t = 0.45\ \[Pi];\), "\n", \(x1 = 2.75;\), "\n", \(x2 = \((Sin[3\ t/2] - x1\ Sin[t/2])\)/ Cos[t/2];\), "\n", \(y1 = \(-2\);\), "\n", \(y2 = \((Sin[3\ t/2] - y1\ Sin[t/2])\)/Cos[t/2];\), "\n", \(x = {x1, x2};\), "\n", \(y = {y1, y2};\), "\n", \(f1[a_] := 2\ Cos[a] + Cos[2\ a]\), "\n", \(f2[a_] := 2\ Sin[a] - Sin[2\ a]\), "\n", \(a = \(-1\);\), "\n", \(b = 1.5;\), "\n", \(p1 = {a\ f1[\[Pi] - t/2] + \((3\ a + 1)\)\ f1[\(-t\)/ 2], a\ f2[\[Pi] - t/2] + \((3\ a + 1)\)\ f2[\(-t\)/ 2]};\), "\n", \(p2 = {b\ f1[\[Pi] - t/2] + \((3\ b + 1)\)\ f1[\(-t\)/2], b\ f2[\[Pi] - t/2] + \((3\ b + 1)\)\ f2[\(-t\)/ 2]};\), "\n", \(c = \(-0.75\);\), "\n", \(d = 1.5;\), "\n", \(q1 = {c\ f1[\(-t\)/2] + \((3\ c + 1)\)\ f1[\[Pi] - t/2], c\ f2[\(-t\)/2] + \((3\ c + 1)\)\ f2[\[Pi] - t/2]};\), "\n", \(q2 = {d\ f1[\(-t\)/2] + \((3\ d + 1)\)\ f1[\[Pi] - t/2], d\ f2[\(-t\)/2] + \((3\ d + 1)\)\ f2[\[Pi] - t/2]};\), "\n", \(lp[ x_, c_] := x\ \((p2 - p1)\) + p1 + {0, c}\), "\n", \(lq[x_, d_] := x\ \((q2 - q1)\) + q1 + {0, d}\), "\n", \(g1 = ParametricPlot[{f1[a], f2[a]}, {a, 0, 2\ \[Pi]}, PlotPoints \[Rule] 300, PlotStyle -> RGBColor[1, 0, 0], AspectRatio \[Rule] Automatic, Ticks \[Rule] None, PlotRange \[Rule] {{\(-3\), 3.5}, {\(-3\), 3}}, PlotStyle \[Rule] AbsoluteThickness[0.8], DisplayFunction \[Rule] Identity];\), "\n", \(g2 = Graphics[{{AbsoluteThickness[0.5], RGBColor[0, 0.5, 0], Line[{{\(-3\), 0}, {3.5, 0}}], Line[{{0, 3}, {0, \(-3\)}}]}, {AbsolutePointSize[5], darkblue, Point[{f1[t], f2[t]}], Point[{f1[\[Pi] - t/2], f2[\[Pi] - t/2]}], Point[{f1[\(-t\)/2], f2[\(-t\)/2]}], Point[{\(-1\)/2\ \((f1[\[Pi] - t/2] + f1[\(-t\)/2])\), \(-1\)/ 2\ \((f2[\[Pi] - t/2] + f2[\(-t\)/2])\)}], Point[{1/2\ \((f1[\[Pi] - t/2] + f1[\(-t\)/2])\), 1/2\ \((f2[\[Pi] - t/2] + f2[\(-t\)/2])\)}]}, {AbsoluteThickness[0.5], darkblue, Line[{x, y}], Line[{p1, p2}], Line[{q1, q2}]}, {AbsoluteThickness[0.5], Dashing[{0.01, 0.01}], RGBColor[1, 0, 0], Circle[{0, 0}, 1]}, {AbsoluteThickness[0.5], darkblue, Dashing[{0.015, 0.015}], Line[{{\(-1\)/2\ \((f1[\[Pi] - t/2] + f1[\(-t\)/2])\), \(-1\)/ 2\ \((f2[\[Pi] - t/2] + f2[\(-t\)/2])\)}, {1/ 2\ \((f1[\[Pi] - t/2] + f1[\(-t\)/2])\), 1/2\ \((f2[\[Pi] - t/2] + f2[\(-t\)/2])\)}}]}}];\), "\n", \(g3 = Show[g1, g2, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\), "\n", \(g4 = ParametricPlot[lp[x, 0.3], {x, 0.164582, 0.189375}, PlotStyle \[Rule] darkblue, DisplayFunction \[Rule] Identity];\), "\n", \(g5 = ParametricPlot[lq[y, \(-0.7\)], {y, 0.152928, 0.17085}, PlotStyle \[Rule] darkblue, DisplayFunction \[Rule] Identity];\), "\n", \(Show[g3, g4, g5, Background \[Rule] lightblue, ImageSize \[Rule] 900, DisplayFunction \[Rule] $DisplayFunction];\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 5.36 (Elimination theory: equations for hypo- and \ epicycloid) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Simplify[ Resultant[x\_1 - 3 + \((2 x\_1 + 6)\) t^2 + \((x\_1 + 1)\) t^4, x\_2 + 2 \( x\_2\) t^2 - 8 t^3 + \(x\_\(\(2\)\(\ \)\)\) t^4, t]/ 4096]\), "\[IndentingNewLine]", \(Simplify[ 4096 % == 2^12 \((\((x\_1^2 + x\_2^2)\)^2 + 8 \((3 \( x\_1\) x\_2^2 - x\_1^3)\) + 18 \((x\_1^2 + x\_2^2)\) - 27)\)]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.38.(iv) (Toroid) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(b = 0.5;\)\), "\n", \(\(t = 0.3\ \[Pi];\)\), "\n", \(II[t_] := 1/Sqrt[b^2\ Cos[t]^2 + Sin[t]^2]\), "\n", \(S1[t_] := Cos[t]\ \((2 + b\ II[t])\)\), "\n", \(S2[t_] := Sin[t]\ \((2\ b + II[t])\)\), "\n", \(T1[t_] := Cos[t]\ \((2 - b\ II[t])\)\), "\n", \(T2[t_] := Sin[t]\ \((2\ b - II[t])\)\), "\n", \(\(p1 = ParametricPlot[{T1[t], T2[t]}, {t, 0, 2\ \[Pi]}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[1]}, AspectRatio \[Rule] Automatic, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = ParametricPlot[{S1[t], S2[t]}, {t, 0, 2\ \[Pi]}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {RGBColor[1, 0, 0], AbsoluteThickness[1]}, AspectRatio \[Rule] Automatic, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = ParametricPlot[{2\ Cos[t], 2\ b\ Sin[t]}, {t, 0, 2\ \[Pi]}, PlotPoints \[Rule] 50, PlotStyle \[Rule] {darkblue, AbsoluteThickness[ .75]}, AspectRatio \[Rule] Automatic, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p4 = Graphics[{AbsoluteThickness[ .75], Dashing[{0.02, 0.02}], {darkblue, Circle[{2\ Cos[t], 2\ b\ Sin[t]}, 1]}, {AbsolutePointSize[5], Point[{2\ Cos[t], 2\ b\ Sin[t]}]}}, AspectRatio \[Rule] Automatic];\)\), "\n", \(\(Show[p1, p2, p3, p4, AxesStyle \[Rule] RGBColor[0, 0.5, 0], ImageSize \[Rule] 600, Background \[Rule] lightblue, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.38.(vi) (Logarithmic spiral) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\n", \(\(\(logspiral[a_, b_]\)[t] = a {\[ExponentialE]^\((b\ t)\) Cos[t], \[ExponentialE]^\((b\ t)\) Sin[t]};\)\), "\n", \(\(p1 = ParametricPlot[ Evaluate[{\(logspiral[1, 0.08]\)[t]}], {t, \(-16\) \[Pi], \[Pi]/3}, PlotStyle \[Rule] RGBColor[1, 0, 0], AspectRatio \[Rule] Automatic, Axes \[Rule] None, PlotRange \[Rule] {{\(-1.25\), 1.25}, {\(-1.25\), 1.25}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = Graphics[{AbsoluteThickness[0.75], {RGBColor[0, 0.5, 0], Line[{{\(-1.1\), 0}, {1.1, 0}}], Line[{{0, 1.1}, {0, \(-1.1\)}}]}}];\)\), "\n", \(\(Show[p1, p2, Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.38.(viii) (Reflected light) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(a = 1;\)\), "\n", \(\(t = 0.2 \[Pi];\)\), "\n", \(\(x1 = a\ Cos[t] - 0.2;\)\), "\n", \(\(p1 = ParametricPlot[{a/2\ Cos[t]\ \((3 - 2\ Cos[t]^2)\), a\ Sin[t]^3}, {t, 0, 2 \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], PlotPoints \[Rule] 50, PlotStyle \[Rule] AbsoluteThickness[1.25], DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = Graphics[{{darkblue, AbsolutePointSize[5], Point[{0, 0}], Point[{a\ Cos[t], a\ Sin[t]}], Point[{a/2\ Cos[t]\ \((3 - 2\ Cos[t]^2)\), a\ Sin[t]^3}]}, {darkblue, AbsoluteThickness[0.75], {Dashing[{0.015, 0.015}], Circle[{0, 0}, a/2], Circle[{3/4\ a\ Cos[t], 3/4\ a\ Sin[t]}, a/4]}, Circle[{0, 0}, a]}, {RGBColor[180/256, 200/256, 0], Line[{{\(-1\)/4, a\ Sin[t]}, {a\ Cos[t], a\ Sin[t]}, {x1, x1\ Tan[2 t] - a\ Cos[t]\ Tan[2 t] + a\ Sin[t]}}]}, {darkblue, Dashing[{0.025, 0.025}], AbsoluteThickness[1], Line[{{a\ Cos[t], a\ Sin[t]}, {0, 0}}]}}, AspectRatio \[Rule] Automatic];\)\), "\n", \(\(Show[p2, p1, Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.39", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex5.39.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 5.41 (H\[ODoubleDot]lder's, Minkowski's and Young's \ inequalities) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\n", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(p = 4;\)\), "\n", \(\(q = 1/\((p - 1)\) + 1;\)\), "\n", \(\(p1 = Plot[x^\((p - 1)\), {x, 0, 1.6}, PlotStyle \[Rule] RGBColor[1, 0, 0], AspectRatio \[Rule] Automatic, Ticks \[Rule] None, PlotRange \[Rule] {{\(-0.2\), 3}, {\(-0.1\), 5.5}}, PlotStyle \[Rule] AbsoluteThickness[1.5], PlotPoints \[Rule] 50, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = Graphics[{Text["\<0\>", {0, 0}], Text["\", {11/7, 0}], Text["\", {2, 0}], Text["\", {0, 4}], Text["\", {0, 2.5}], Text["\", {2, 4}], {Table[{darkblue, Line[{{n/7.32, 0}, {n/7.32, \((n/7.32)\)^\((p - 1)\)}}]}, {n, 1, 11}], Table[{darkblue, Line[{{0, n/4}, {\((n/4)\)^\((q - 1)\), n/4}}]}, {n, 1, 10}]}, {AbsoluteThickness[2], {darkblue, Line[{{11/7.32, 0}, {11/7.32, 2.5}}]}, {darkblue, Line[{{0, 2.5}, {11/7.32, 2.5}}]}}}];\)\), "\n", \(\(Show[p1, p2, PlotRange \[Rule] {{\(-0.2\), 2.25}, {\(-0.1\), 4.5}}, AspectRatio \[Rule] Automatic, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 300, DisplayFunction \[Rule] $DisplayFunction, DefaultFont \[Rule] 12. ];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.46 (Diameter of submanifold) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\n", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(r[t_] := 5\ Sqrt[t]\), "\n", \(s[t_] := 2 t + 1\), "\n", \(f1[t_] := r[t]\ Cos[t]\), "\n", \(f2[t_] := s[t]\ Sin[t] + t - 10\), "\n", \(\(g1[t_] = \(f1'\)[t];\)\), "\n", \(\(g2[t_] = \(f2'\)[t];\)\), "\n", \(\(y = 5.17984;\)\), "\n", \(\(x = 8.33324;\)\), "\n", \(\(ay = 1;\)\), "\n", \(\(ax = 0.75;\)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[{f1[t], f2[t]}, {t, 0.2\ \[Pi], 2.75\ \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], PlotPoints \[Rule] 100, PlotRange \[Rule] {{\(-20\), 20}, {\(-20\), 20}}, AspectRatio \[Rule] 1, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = Graphics[{{AbsolutePointSize[4], darkblue, Point[{f1[x], f2[x]}], Point[{f1[y], f2[y]}]}, Text["\", {f1[x], f2[x] + 1}], Text["\", {f1[y], f2[y] - 1}], {darkblue, Line[{{f1[x] - ax\ g1[x], f2[x] - ax\ g2[x]}, {f1[x] + ax\ g1[x], f2[x] + ax\ g2[x]}}]}, {darkblue, Line[{{f1[y] - ay\ g1[y], f2[y] - ay\ g2[y]}, {f1[y] + ay\ g1[y], f2[y] + ay\ g2[y]}}]}, {Dashing[{0.02, 0.02}], darkblue, Line[{{f1[x], f2[x]}, {f1[y], f2[y]}}]}}];\)\), "\n", \(\(Show[p1, p2, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 5.51 (Tractrix and pseudosphere) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is unable to directly evaluate the integral in part (i), although it is \ able to find an antiderivative for the integrand." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Plot::plnr]\), "\n", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(int[x_] := Integrate[Sqrt[1 - t^2]/t, {t, x, 1}, Assumptions \[Rule] {0 < x < 1}]\), "\[IndentingNewLine]", \(FullSimplify[int[x], {0 < x < 1}]\ \), "\[IndentingNewLine]", \(\(i[t_] = Integrate[Sqrt[1 - t^2]/t, t];\)\), "\[IndentingNewLine]", \(f[x_] = i[1] - i[x]\), "\[IndentingNewLine]", \(\(\(Plot[f[x], {x, 0, 1}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Ticks \[Rule] None, AspectRatio \[Rule] 1, PlotRange \[Rule] {\(-0.2\), 10}, Background \[Rule] lightblue, ImageSize \[Rule] 500];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Simplify[\(({x, \(-f[x]\)} /. x \[Rule] Sin[s])\), 0 < s < \[Pi]/2]\), "\[IndentingNewLine]", \(\(\(% /. s \[Rule] \[Theta] + \[Pi]/2\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(tractrix[s_] := {Sin[s], Cos[s] + Log[Tan[s/2]]}\), "\[IndentingNewLine]", \(\[CurlyPhi][s_, t_] := {Sin[s]\ Cos[t], Sin[s]\ Sin[t], Cos[s] + Log[Tan[s/2]]}\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Transpose[{\[CurlyPhi][s, t]}]\), "\[IndentingNewLine]", \(Simplify[ Transpose[{\[PartialD]\_s\ \[CurlyPhi][s, t]}]]\), "\[IndentingNewLine]", \(Transpose[{\[PartialD]\_t\ \[CurlyPhi][s, t]}]\), "\[IndentingNewLine]", \(\(\(Transpose[{Simplify[ 1/Cos[s]\ Cross[\[PartialD]\_s\ \[CurlyPhi][s, t], \[PartialD]\_t\ \[CurlyPhi][s, t]]]}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[tractrix[s], {s, 0.005\ \[Pi], 0.9925\ \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRegion \[Rule] {{0.2, 0.8}, {0.2, 0.8}}, PlotPoints \[Rule] 100, AspectRatio \[Rule] Automatic, Ticks \[Rule] None, PlotRange \[Rule] {{\(-0.5\), 1.5}, {\(-2\), 2}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = ParametricPlot3D[\[CurlyPhi][s, t] // Evaluate, {s, 0.001, \[Pi]}, {t, 0, 2 \[Pi]}, Axes \[Rule] None, ViewPoint \[Rule] {1, \(-2\), 1/2}, Boxed \[Rule] False, PlotPoints \[Rule] {80, 60}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[GraphicsArray[{p1, p2}], Background \[Rule] lightblue, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Exercise 5.58 (Exponential of antisymmetric matrix is special orthogonal \ in ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "3"], TraditionalForm]]], ") " }], "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(r[{a1_, a2_, a3_}] := {{0, \(-a3\), a2}, {a3, 0, \(-a1\)}, {\(-a2\), a1, 0}};\)\), "\[IndentingNewLine]", \(R[p0_, p_List] := \((p0^2 - p[\([1]\)]^2 - p[\([2]\)]^2 - p[\([3]\)]^2)\) IdentityMatrix[3] + 2 \((Transpose[{p}] . {p} + p0*r[p])\)\), "\[IndentingNewLine]", \(co[p_, n_] := Table[Subscript[p, j], {j, n}]\), "\[IndentingNewLine]", \(\(\(pv = co[p, 3];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(MatrixForm[R[p\_0, pv]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(\(MatrixForm[ Simplify[Transpose[R[p0, pv]] . R[p0, pv]] /. p0^2 + pv[\([1]\)]^2 + pv[\([2]\)]^2 + pv[\([3]\)]^2 \[Rule] 1]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(av = co[a, 3];\)\), "\[IndentingNewLine]", \(\(p0 = Cos[\[Alpha]\/2];\)\), "\[IndentingNewLine]", \(\(pv = Sin[\[Alpha]\/2] av;\)\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(\(\(MatrixForm[ Simplify[TrigReduce[R[p0, pv]] /. av[\([1]\)]^2 \[Rule] 1 - av[\([2]\)]^2 - av[\([3]\)]^2]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(angle[a_List] := ArcCos[\((\(-1\) + a[\([1, 1]\)] + a[\([2, 2]\)] + a[\([3, 3]\)])\)/ 2]\), "\[IndentingNewLine]", \(ax[a_List] := Simplify[\((a - Transpose[a])\)/\((2 Sin[angle[a]])\)]\), "\[IndentingNewLine]", \(\(\(axis[ a_List] := {\(ax[a]\)[\([3, 2]\)], \(-\(ax[a]\)[\([3, 1]\)]\), \(ax[ a]\)[\([2, 1]\)]}\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(0^2 + 0^2 + 3^2 + 4^2 \[Equal] 5^2\), "\[IndentingNewLine]", \(\(rot = R[3/5, {4/5, 0, 0}];\)\), "\[IndentingNewLine]", \(Simplify[1/25 MatrixForm[25 rot]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(180/\[Pi]\ N[angle[rot]]\), "\[IndentingNewLine]", \(axis[rot]\), "\[IndentingNewLine]", \(Simplify[Eigensystem[rot]]\), "\[IndentingNewLine]", \(\(\(Expand[ 25 Det[rot - x\ IdentityMatrix[3]]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(1^2 + 1^2 + 3^2 + 5^2 \[Equal] 6^2\), "\[IndentingNewLine]", \(\(rot = R[1/6, {5/6, 1/2, 1/6}];\)\), "\[IndentingNewLine]", \(Simplify[1/36 MatrixForm[36 rot]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(180/\[Pi]\ N[angle[rot]]\), "\[IndentingNewLine]", \(Sqrt[35] axis[rot]\), "\[IndentingNewLine]", \(Simplify[Eigensystem[rot]]\), "\[IndentingNewLine]", \(\(\(Expand[9 Det[rot - x\ IdentityMatrix[3]]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(2^2 + 2^2 + 4^2 + 5^2 \[Equal] 7^2\), "\[IndentingNewLine]", \(\(rot = R[5/7, {\(-2\)/7, 4/7, 2/7}];\)\), "\[IndentingNewLine]", \(Simplify[1/49\ MatrixForm[49 rot]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(180/\[Pi]\ N[angle[rot]]\), "\[IndentingNewLine]", \(Sqrt[6] axis[rot]\), "\[IndentingNewLine]", \(Simplify[Eigensystem[rot]]\), "\[IndentingNewLine]", \(\(\(Expand[ 49 Det[rot - x\ IdentityMatrix[3]]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(1^2 + 2^2 + 4^2 + 10^2 \[Equal] 11^2\), "\[IndentingNewLine]", \(\(rot = R[4/11, {\(-2\)/11, 1/11, \(-10\)/11}];\)\), "\[IndentingNewLine]", \(Simplify[1/121\ MatrixForm[121 rot]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(180/\[Pi]\ N[angle[rot]]\), "\[IndentingNewLine]", \(Sqrt[105] axis[rot]\), "\[IndentingNewLine]", \(Simplify[Eigensystem[rot]]\), "\[IndentingNewLine]", \(Expand[121 Det[rot - x\ IdentityMatrix[3]]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 5.66 (Reflections, rotations, Cayley-Klein parameters and \ Sylvester's Theorem) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["This is an illustration of ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], "Sylvester's Theorem from part (iii).", StyleBox[" Text has to be added using Xfig. ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]] }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["The code below might be improved.", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(Off[General::spell];\)\), "\n", \(\(Off[General::spell1];\)\[IndentingNewLine]\[IndentingNewLine] (*\ controlling\ the\ output\ of\ the\ graphics\ *) \[IndentingNewLine]\), \ "\n", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(raster = 0.035;\)\), "\n", \(\(point = 0.001;\)\), "\n", \(\(thick = 0.002;\)\), "\n", \(\(dash = 0.005;\)\[IndentingNewLine]\[IndentingNewLine] (*\ polar\ coordinates\ $ \((\(\(\\\)\(alpha\)\), \(\(\\\)\(theta\)\))\) $\ \ of\ unit\ vectors\ $a_ 1 $\ and\ $a_ 2 $\ and\ angles\ $\\alpha_ 1 $\ and\ $\\alpha_ 2 $\ of\ rotation\ about\ $p$\ and\ $q$, respectively, \ all\ in\ multiples\ of\ $\\pi$\ *) \n\), "\[IndentingNewLine]", \(\(ap = 4/5;\)\), "\n", \(\(tp = 1/10;\)\), "\n", \(\(aq = 1/10;\)\), "\n", \(\(tq = 1/3;\)\), "\n", \(\(anglep = 1/2;\)\), "\n", \(\(angleq = 2/3;\)\[IndentingNewLine]\n (*\ stripping\ of\ excess\ parentheses\ and\ definition\ of\ norm\ of\ \ vector\ for\ purposes\ of\ computation\ of\ length\ of\ vector\ *) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(<< LinearAlgebra`Orthogonalization`;\)\), "\n", \(\(Strip[a_] := Apply[Sequence, a];\)\), "\n", \(\(norm[a1_, a2_, a3_] := N[Sqrt[\((a1)\)^2 + \((a2)\)^2 + \((a3)\)^2]];\)\), "\n", \(\(normS[a_] := norm[Strip[a]];\)\n\[IndentingNewLine] (*\ transparant\ unit\ sphere\ *) \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(x[1, a_, t_] := Cos[a\ \[Pi]] Cos[t\ \[Pi]];\)\), "\n", \(\(x[2, a_, t_] := Sin[a\ \[Pi]] Cos[t\ \[Pi]];\)\), "\n", \(\(x[3, a_, t_] := Sin[t\ \[Pi]];\)\), "\n", \(\(x[a_, t_] := Table[x[j, a, t], {j, 3}];\)\), "\n", \(\(list = Table[Point[x[a, t]], {a, \(-\[Pi]\), \[Pi], raster}, {t, \(-\[Pi]\)/2, \[Pi]/2, raster}];\)\), "\n", \(\(S = Graphics3D[{PointSize[point], list}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False];\)\n\[IndentingNewLine] (*\ two\ unit\ vectors\ $a_ 1 $\ and\ $a_ 2 $\ that\ determine\ the\ axes\ of\ rotation\ $\\R\ p$\ and\ $\\R\ \ q$\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(a[1] = \(a1 = x[ap, tp]\);\)\), "\n", \(\(a[2] = \(a2 = x[aq, tq]\);\)\[IndentingNewLine]\n (*\ solid\ lines\ that\ give\ the\ vectors\ $a1$\ and\ $a2$\ *) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(LP = Graphics3D[{RGBColor[1, 0, 0], Thickness[thick], Line[{0*a1, a1}]}];\)\), "\n", \(\(LQ = Graphics3D[{RGBColor[1, 0, 0], Thickness[thick], Line[{0*a2, a2}]}];\)\n\[IndentingNewLine] (*\ $x_ 2 $\ is\ normalized\ cross\ product\ of\ $q$\ and\ $p$\ *) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(x[2] = \(x2 = Normalize[ Cross[a2, a1]]\);\)\[IndentingNewLine]\[IndentingNewLine] (*\ dashed\ line\ that\ gives\ the\ vector\ $x_ 2 $\ *) \[IndentingNewLine]\), "\n", \(\(LX2 = Graphics3D[{Thickness[1.75*thick], Dashing[{dash, 2*dash}], Line[{0*x2, x2}]}];\)\[IndentingNewLine]\[IndentingNewLine] (*\ matrix\ $R_ {a, \((a_ 1, a_ 2, a_ 3)\)} $\ of\ rotation\ by\ angle\ $a$\ about\ axis\ $ \((a_ 1, a_ 2, a_ 3)\) $\ *) \n\), "\[IndentingNewLine]", \(\(r[a1_, a2_, a3_] := {{0, \(-a3\), a2}, {a3, 0, \(-a1\)}, {\(-a2\), a1, 0}};\)\), "\n", \(\(R[a_, a1_, a2_, a3_] := Cos[a] IdentityMatrix[3] + \((1 - Cos[a])\) Transpose[{{a1, a2, a3}}] . {{a1, a2, a3}} + Sin[a] r[a1, a2, a3];\)\[IndentingNewLine]\[IndentingNewLine] (*\ point\ on\ geodesic\ issuing\ from\ point\ $x_ 2 $\ arising\ as\ image\ under\ rotation\ by\ angle\ $a$\ about\ \ axis\ $ \((a_ 1, a_ 2, a_ 3)\) $\ *) \[IndentingNewLine]\), "\n", \(\(geod[a_, a1_, a2_, a3_] := R[a, a1, a2, a3] . x2;\)\[IndentingNewLine]\n (*\ geodesics\ obtained\ by\ rotating\ $x_ 2 $\ by\ the\ angle\ $a$\ about\ the\ axis\ $p$\ and\ $q$, \ \(\(resp\)\(.\)\)\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(geodp[a_] := R[a, Strip[a1]] . x2;\)\), "\n", \(\(geodq[a_] := R[a, Strip[a2]] . x2;\)\n\[IndentingNewLine] (*\ graphics\ of\ the\ segment\ of\ the\ geodesics\ from\ $x_ 2 $\ to\ $x_ 1 $\ and\ $x_ 3 $\ resp . \ determined\ by\ angle\ $a$\ running\ between\ $0$\ and\ $s$\ *) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(GEODP[s_] := ParametricPlot3D[Evaluate[geodp[a]], {a, 0, s}, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(GEODQ[s_] := ParametricPlot3D[Evaluate[geodq[a]], {a, 0, s}, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(GEODX2X1 = GEODP[\(-anglep\)*\[Pi]/2];\)\), "\n", \(\(\(GEODX2X3 = GEODQ[angleq*\[Pi]/2];\)\(\n\) \)\), "\[IndentingNewLine]", \(\(x[1] = \(x1 = geodp[\(-anglep\)*\[Pi]/2]\);\)\), "\n", \(\(x[3] = \(x3 = geodq[angleq*\[Pi]/2]\);\)\[IndentingNewLine]\n (*\ dashed\ lines\ that\ give\ the\ vectors\ $x_ 1 $\ and\ $x_ 3 $\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(LX1 = Graphics3D[{Thickness[thick], Dashing[{dash, 2 dash}], Line[{0*x1, x1}]}];\)\), "\n", \(\(LX3 = Graphics3D[{Thickness[thick], Dashing[{dash, 2 dash}], Line[{0*x3, x3}]}];\)\n\[IndentingNewLine] (*\ axis\ $a_ 3 $\ that\ is\ the\ normalized\ cross\ product\ of\ $x_ 1 $\ and\ $x_ 3 $\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(a[3] = \(a3 = Normalize[ Cross[x1, x3]]\);\)\[IndentingNewLine]\[IndentingNewLine] (*\ solid\ line\ that\ gives\ the\ axis\ of\ rotation\ $a3$\ *) \ \[IndentingNewLine]\), "\n", \(\(LR = Graphics3D[{RGBColor[1, 0, 0], Thickness[2 thick], Line[{0*a3, a3}]}];\)\[IndentingNewLine]\[IndentingNewLine] (*\ geodesic\ connecting\ $x_ 1 $\ and\ $x_ 3 $\ *) \n\), "\[IndentingNewLine]", \(\(geodr[a_] := R[a, Strip[a3]] . x1;\)\), "\n", \(\(GEODR[s_] := ParametricPlot3D[Evaluate[geodr[a]], {a, 0, s}, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(angler = 2 ArcCos[x1 . x3];\)\), "\n", \(\(\(GEODX1X3 = GEODR[angler/2];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(p0 = Cos[anglep*\[Pi]/2];\)\), "\n", \(\(q0 = Cos[angleq*\[Pi]/2];\)\), "\n", \(\(p = Sin[anglep*\[Pi]/2]\ a1;\)\), "\n", \(\(q = Sin[angleq*\[Pi]/2]\ a2;\)\), "\n", \(\(\(r = q0\ p + p0\ q + Cross[q, p];\)\(\[IndentingNewLine]\) \)\), "\n", \(\(P = Graphics3D[{PointSize[7 point], Point[p]}, Shading \[Rule] False];\)\), "\n", \(\(Q = Graphics3D[{PointSize[7 point], Point[q]}, Shading \[Rule] False];\)\), "\n", \(\(\(R = Graphics3D[{PointSize[10 point], Point[r]}, Shading \[Rule] False];\)\(\[IndentingNewLine]\) \)\), "\n", \(\({A1, A2, A3} = Table[Graphics3D[{PointSize[7 point], Point[a[j]]}, Shading \[Rule] False], {j, 3}];\)\), "\n", \(\({X1, X2, X3} = Table[Graphics3D[{PointSize[7 point], Point[x[j]]}, Shading \[Rule] False], {j, 3}];\)\n\[IndentingNewLine] (*\ final\ illustration\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(f = 2.2; view = f*Normalize[x1 + x2 + x3] + {0, 0, 1};\), "\[IndentingNewLine]", \(\(Show[S, LP, LQ, LR, LX1, LX2, LX3, GEODX2X1, GEODX2X3, GEODX1X3 /. Line[a_List] \[Rule] {Thickness[ .005], Line[a]}, P, Q, R, A1, A2, A3, X1, X2, X3, Axes \[Rule] False, PlotRange \[Rule] All, BoxRatios \[Rule] {1, 1, 1}, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] view, ImageSize \[Rule] 900, DefaultColor \[Rule] darkblue, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\[IndentingNewLine]\ \[IndentingNewLine] (*\ verification\ *) \[IndentingNewLine]\), "\n", \(N[Normalize[r]] - Map[N, a3]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["NEW EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["New Exercise 0.0 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Example of a bounded continuous function ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox["f", FontSlant->"Italic", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]], StyleBox[" on an open interval that assumes its maximum in the interior of \ the interval and has a nowhere vanishing derivative. ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]] }], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Solve::verif]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(Plot[ArcSin[y], {y, \(-1\), 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle -> RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(f[x_] := ArcSin[2\ Sqrt[x]/\((x + 1)\)]\), "\[IndentingNewLine]", \(f[1]\), "\[IndentingNewLine]", \(\(Plot[f[x], {x, 0, 4}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle -> RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(g[x_] = \(f'\)[x]\), "\[IndentingNewLine]", \(FullSimplify[ ComplexExpand[g[x]], {Im[x + 1] \[Equal] 0, x + 1 > 0}]\), "\[IndentingNewLine]", \(FullSimplify[% \[Equal] \(-Sign[x - 1]\)/\((\((x + 1)\) Sqrt[x])\), Im[x] \[Equal] 0]\), "\[IndentingNewLine]", \(Limit[g[x], x \[Rule] 1, Direction \[Rule] 1]\), "\[IndentingNewLine]", \(Limit[g[x], x \[Rule] 1, Direction \[Rule] \(-1\)]\), "\[IndentingNewLine]", \(\(Plot[g[x], {x, 0.0001, 4}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle -> RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(Solve[g[x] \[Equal] 0, x]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["New Exercise 4.0 (Lissajous figures) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[{Sin[t], Sin[2 t]}, {t, 0, 8 \[Pi]}, AspectRatio \[Rule] 1, PlotPoints \[Rule] 2000, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[{Sin[2 t], Sin[3 t]}, {t, 0, 8 \[Pi]}, AspectRatio \[Rule] 1, PlotPoints \[Rule] 2000, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[GraphicsArray[{p1, p2}], DisplayFunction -> $DisplayFunction];\)\), "\[IndentingNewLine]", \(\(p3 = ParametricPlot[{Sin[2 t], Sin[5 t]}, {t, 0, 8 \[Pi]}, AspectRatio \[Rule] 1, PlotPoints \[Rule] 2000, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot[{Sin[7 t], Sin[23 t]}, {t, 0, 8 \[Pi]}, AspectRatio \[Rule] 1, PlotPoints \[Rule] 2000, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[GraphicsArray[{p3, p4}], DisplayFunction -> $DisplayFunction];\)\), "\[IndentingNewLine]", \(\(ParametricPlot[{Cos[33 t + 0.2 \[Pi]], Sin[61 t + 19/27 \[Pi]]}, {t, 0, 8 \[Pi]}, AspectRatio \[Rule] 1, PlotPoints \[Rule] 2000, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 800];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ New Exercise 5.0 (Quadratic equation from geometric point of \ view) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[ General::spell1]\), "\[IndentingNewLine]", \(Off[ N::meprec]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\^2\)+2\!\(\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\)\_1\)\!\(\* StyleBox[\"x\",\nFontSlant->\"Italic\"]\)\>\"", "]"}], "\[IndentingNewLine]", \(A = {{1, 1}, {1, 0}}\), "\[IndentingNewLine]", \(\[Tau]\_\(\_+\) = \(Eigenvalues[ A]\)[\([1]\)]\), "\[IndentingNewLine]", \(\[Tau]\_\(\_-\) = \ \(Eigenvalues[ A]\)[\([2]\)]\), "\[IndentingNewLine]", \ \(FullSimplify[\[Tau]\_\(\_+\) == GoldenRatio]\), "\[IndentingNewLine]", \(FullSimplify[ 1/\[Tau]\_\(\_+\) \[Equal] \(-\[Tau]\_\(\_-\)\)]\), \ "\[IndentingNewLine]", \(a1 = \(Eigenvectors[ A]\)[\([1]\)]\), "\[IndentingNewLine]", \(a2 = \(-\(Eigenvectors[ A]\)[\([2]\)]\)\), "\[IndentingNewLine]", \(FullSimplify[ Norm[a1] == PowerExpand[ 5^\((1/4)\) Sqrt[\[Tau]\_\(\_+\)]]]\), "\[IndentingNewLine]", \(FullSimplify[ Norm[a2] == PowerExpand[ 5^\((1/4)\) Sqrt[\(-\[Tau]\_\(\_-\)\)]]]\), "\[IndentingNewLine]", \(Orth = 1/\((5^\((1/4)\))\) Transpose[{{Sqrt[\[Tau]\_\(\_+\)], Sqrt[\(-\[Tau]\_\(\_-\)\)]}, {Sqrt[\(-\[Tau]\_\(\_-\)\)], \ \(-Sqrt[\[Tau]\_\(\_+\)]\)}}]\), "\[IndentingNewLine]", \(Simplify[ Transpose[Orth] . Orth]\), "\[IndentingNewLine]", \(Simplify[ Inverse[Orth] . A . Orth]\), "\[IndentingNewLine]", \(Simplify[ Orth . A . Orth]\), "\[IndentingNewLine]", \(z = \({z1, z2} = Orth . {x, y\_1}\);\), "\[IndentingNewLine]", RowBox[{\(Expand[ Simplify[\(\[Tau]\_\(\_+\)\) z1^2 + \(\[Tau]\_\(\_-\)\) z2^2]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(<< Graphics`ContourPlot3D`\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\n", \(p = ContourPlot3D[ x^2 + 2 y1\ x + y2, {x, \(-4\), 4}, {y1, \(-4\), 4}, {y2, \(-4\), 4}, PlotPoints \[Rule] 3 {3, 4}, PlotRange \[Rule] {{\(-3\), 3}, {\(-2\), 2}, {\(-2\), 2}}, ViewPoint \[Rule] {2, 1.4, 1.6}, \[IndentingNewLine]ContourStyle \[Rule] RGBColor[0.7, 0.7, 1], LightSources \[Rule] {{{1, 0, 0}, RGBColor[0, 0, 1]}, {{0, 1, 0}, RGBColor[0, 0, 1]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, Boxed \[Rule] False, ImageSize \[Rule] 700, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(thick = 0.004;\), "\n", \(x = Graphics3D[{Thickness[thick], {RGBColor[1, 0, 0], Line[{{0, 0, 0}, {6, 0, 0}}]}}];\), "\n", \(y1 = Graphics3D[{Thickness[thick], {RGBColor[1, 0, 0], Line[{{0, 0, 0}, {0, 3, 0}}]}}];\), "\n", \(y2 = Graphics3D[{Thickness[thick], {RGBColor[1, 0, 0], Line[{{0, 0, 0}, {0, 0, 1}}]}}];\), "\[IndentingNewLine]", RowBox[{ RowBox[{"t1", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\* StyleBox[\"x\",\nFontSlant->\"Italic\"]\)-axis\>\"", ",", \({4.5, 0, 0}\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"t2", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\)\_1\)-axis\>\"", ",", \({0, 2.5, 0}\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"t3", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\)\_2\)-axis\>\"", ",", \({0, 0, 1.2}\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{\(Show[p, x, y1, y2, t1, t2, t3, TextStyle \[Rule] {FontFamily -> "\", FontSize \[Rule] 14. }, DisplayFunction \[Rule] $DisplayFunction];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Off[ General::spell1]\), "\n", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\n", RowBox[{\(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\n"}], "\[IndentingNewLine]", \(xmax = 2.5;\), "\n", \(xmin = \(-xmax\);\), "\n", \(y1max = xmax;\), "\n", \(y1min = \(-y1max\);\), "\n", \(y1maxc = xmax;\), "\n", \(y1minc = \(-y1maxc\);\), "\n", \(planex = \ \(-13\);\), "\n", \(planey2 = \(-28\);\), "\n", \(view = {4, 2, 6};\), "\n", \(raster = 0.05;\), "\n", \(point = 0.001;\), "\n", RowBox[{\(dash = 0.01;\), "\n"}], "\n", \(f[x_, y1_] := \(-x^2\) - 2 x\ y1\), "\n", \(h[ y1_] := \(-y1\)\), "\n", \(xboundaries = ParametricPlot3D[{\[IndentingNewLine]{x, y1min, f[x, y1min]}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ left\ border\ of\ surface\ *) \[IndentingNewLine]{x, y1max, f[x, y1max]}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ right\ border\ of\ surface\ *) \[IndentingNewLine]{x, y1min, planey2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ left\ line\ in\ bottom\ plane\ *) \ \[IndentingNewLine]{x, y1max, planey2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ right\ line\ in\ bottom\ plane\ *) \[IndentingNewLine]{planex, y1min, f[x, y1min]}, \ \ \ (*\ projection\ of\ left\ border\ of\ surface\ on\ back\ plane\ \ *) \[IndentingNewLine]{planex, y1max, f[x, y1max]}}, \ (*\ projection\ of\ right\ border\ of\ surface\ on\ back\ plane\ \ *) \[IndentingNewLine]{x, xmin, xmax}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\), "\n", \(y1boundaries = ParametricPlot3D[{\[IndentingNewLine]{xmax, y1, f[xmax, y1]}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ front\ border\ of\ surface\ *) \[IndentingNewLine]{xmin, y1, f[xmin, y1]}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ back\ border\ of\ surface\ *) \[IndentingNewLine]{planex, y1, f[xmax, y1]}, \ \ \ \ \ \ \ \ \ (*\ projection\ of\ front\ border\ of\ surface\ on\ back\ plane\ \ *) \[IndentingNewLine]{planex, y1, f[xmin, y1]}, \ \ \ \ \ \ \ \ \ (*\ projection\ of\ back\ border\ of\ surface\ on\ back\ plane\ \ *) \[IndentingNewLine]{xmax, y1, planey2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ front\ line\ in\ bottom\ plane\ *) \[IndentingNewLine]{xmin, y1, planey2}}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ back\ line\ in\ bottom\ plane\ *) \ \[IndentingNewLine]{y1, y1min, y1max}, ViewPoint \[Rule] view, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\), "\n", \(crit = ParametricPlot3D[{\[IndentingNewLine]{h[y1], y1, planey2}, \[IndentingNewLine]{h[y1], y1, f[h[y1], y1]}, \[IndentingNewLine]{planex, h[y1], f[h[y1], y1]}\[IndentingNewLine]}, {y1, y1minc, y1maxc}, ViewPoint \[Rule] view, Axes \[Rule] False, ViewPoint \[Rule] view, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\), "\n", \(planes = Graphics3D[{Dashing[{dash, dash}], \[IndentingNewLine]Line[{{xmin, y1min, planey2}, {planex, y1min, planey2}}], \[IndentingNewLine]Line[{{xmin, y1max, planey2}, {planex, y1max, planey2}}], \[IndentingNewLine]Line[{{planex, y1min, planey2}, {planex, y1max, planey2}}], \[IndentingNewLine]Line[{{planex, y1min, planey2}, {planex, y1min, 4}}], \[IndentingNewLine]Line[{{planex, y1max, planey2}, {planex, y1max, 4}}]}, \[IndentingNewLine]Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\), "\n", \(list1 = Table[{darkblue, Point[{x, y1, f[x, y1]}]}, {x, xmin, xmax, raster}, {y1, y1min, y1max, raster}];\), "\n", \(list2 = Table[{darkblue, Point[{x, y1, planey2}]}, {x, xmin, xmax, raster}, {y1, y1min, y1max, raster}];\), "\n", \(list3 = Table[{darkblue, Point[{planex, y1, f[x, y1]}]}, {x, xmin, xmax, raster}, {y1, y1min, y1max, raster}];\), "\n", \(cloud = Graphics3D[{PointSize[point], list1, list2, list3}, Axes \[Rule] False, PlotRange \[Rule] {{xmin, xmax}, {y1min, y1max}, {\(-10\), 16}}, AspectRatio \[Rule] 1, Boxed \[Rule] False, ViewPoint \[Rule] {viewx, viewy1, viewy2}, DisplayFunction \[Rule] Identity];\), "\n", \(t1 = Graphics3D[ Text["\<\!\(\*StyleBox[\"S\",\nFontSlant->\"Italic\"]\)\>", {xmax, y1min - 0.5, planey2 - 0.4}]];\), "\n", \(t2 = Graphics3D[ Text["\<\[CapitalSigma]\>", {0, y1min - 1.8, 2.2}]];\), "\n", \(t3 = Graphics3D[ Text["\<\!\(\*StyleBox[\"P\",\nFontSlant->\"Italic\"]\)\>", \ {planex, 0, 1.5}]];\), "\n", \(Show[cloud, xboundaries, y1boundaries, crit, planes, t1, t2, t3, TextStyle \[Rule] {FontFamily -> "\", FontSize \[Rule] 14. }, ViewPoint \[Rule] view, PlotRange \[Rule] All, DefaultColor \[Rule] RGBColor[1, 0, 0], Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)}], "Input"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(thick = 0.003;\)\), "\n", \(\(l = Graphics3D[{Thickness[thick], RGBColor[0, 0.5, 0], Line[{{3, 0, 0}, {0, 0, 0}, {0, 2, 0}}], Line[{{0, 0, 0}, {0, 0, 1}}]}];\)\), "\[IndentingNewLine]", \(\(view = {1, 1.5, 2.5};\)\), "\n", \(\(plotrange = {{\(-5\), 5}, {\(-4\), 4}, {\(-4.5\), 4.5}};\)\), "\n", \(\(raster = 0.02;\)\), "\[IndentingNewLine]", \(\(point = 0.001;\)\), "\n", \(\(list = Table[{darkblue, Point[{x, y1, \(-x^2\) - 2 y1*x}]}, {x, \(-6\), 6, raster}, {y1, \(-8\), 8, raster}];\)\), "\n", \(\(g1 = Graphics3D[{PointSize[point], list}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(g2 = ParametricPlot3D[ Append[{\(-y1\), y1, y1^2}, RGBColor[1, 0, 0]] // Evaluate, {y1, \(-8\), 8}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(g3 = ParametricPlot3D[ Append[{1.5, \(-1.5\) - y, \((1.5)\)^2 + 3*y}, RGBColor[1, 0, 0]] // Evaluate, {y, \(-8\), 8}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[l, g1, g2, g3, Axes \[Rule] False, Boxed \[Rule] False, BoxRatios \[Rule] {1, 1, 2}, ViewPoint \[Rule] view, PlotRange \[Rule] plotrange, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 6: THEORY", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["\<\ Proposition 6.1.2 Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(Graphics[{RGBColor[1, 0, 0], AbsolutePointSize[4.5], Point[{1, 0}], Point[{2.5, 0}], Point[{4, 0}], Point[{5.5, 0}], Point[{6.5, 0}], Point[{8, 0}], Point[{8.5, 0}], Point[{9.5, 0}], Point[{10.5, 0}], Point[{12, 0}], Point[{14, 0}], Point[{0, 2}], Point[{0, 3.5}], Point[{0, 4.5}], Point[{0, 5.5}], Point[{0, 7}], Point[{0, 8}], Point[{0, 9.5}], Point[{0, 10.5}], Point[{0, 11}], Point[{0, 13}], Point[{0, 14.5}], Point[{0, 16}]}];\)\), "\n", \(\(Graphics[{AbsoluteThickness[0.5], Dashing[{0.0125, 0.0125}], Line[{{1, 2}, {14, 2}}], Line[{{1, 3.5}, {14, 3.5}}], Line[{{1, 4.5}, {14, 4.5}}], Line[{{1, 5.5}, {14, 5.5}}], Line[{{1, 7}, {14, 7}}], Line[{{1, 8}, {14, 8}}], Line[{{1, 9.5}, {14, 9.5}}], Line[{{1, 10.5}, {14, 10.5}}], Line[{{1, 11}, {14, 11}}], Line[{{1, 13}, {14, 13}}], Line[{{1, 14.5}, {14, 14.5}}], Line[{{1, 16}, {14, 16}}], Line[{{1, 2}, {1, 16}}], Line[{{2.5, 2}, {2.5, 16}}], Line[{{4, 2}, {4, 16}}], Line[{{5.5, 2}, {5.5, 16}}], Line[{{6.5, 2}, {6.5, 16}}], Line[{{8, 2}, {8, 16}}], Line[{{8.5, 2}, {8.5, 16}}], Line[{{9.5, 2}, {9.5, 16}}], Line[{{10.5, 2}, {10.5, 16}}], Line[{{12, 2}, {12, 16}}], Line[{{14, 2}, {14, 16}}]}];\)\), "\n", \(\(Graphics[{AbsoluteThickness[2], Line[{{1, 2}, {14, 2}}], Line[{{1, 16}, {14, 16}}], Line[{{1, 2}, {1, 16}}], Line[{{14, 2}, {14, 16}}], Line[{{4, 2}, {4, 5.5}}], Line[{{5.5, 2}, {5.5, 16}}], Line[{{10.5, 2}, {10.5, 8}}], Line[{{6.5, 8}, {6.5, 9.5}}], Line[{{8, 8}, {8, 14.5}}], Line[{{9.5, 8}, {9.5, 16}}], Line[{{10.5, 2}, {10.5, 8}}], Line[{{12, 11}, {12, 16}}], Line[{{5.5, 3.5}, {10.5, 3.5}}], Line[{{5.5, 4.5}, {10.5, 4.5}}], Line[{{1, 5.5}, {5.5, 5.5}}], Line[{{10.5, 7}, {14, 7}}], Line[{{5.5, 8}, {14, 8}}], Line[{{5.5, 9.5}, {8, 9.5}}], Line[{{2.5, 10.5}, {5.5, 10.5}}], Line[{{9.5, 11}, {14, 11}}], Line[{{1, 13}, {5.5, 13}}], Line[{{5.5, 14.5}, {9.5, 14.5}}], Line[{{2.5, 5.5}, {2.5, 13}}]}];\)\), "\n", \(\(Show[%, %%, %%%, Background \[Rule] lightblue, DefaultColor \[Rule] darkblue, ImageSize \[Rule] 600, Axes \[Rule] True, Ticks \[Rule] None];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Theorem 6.2.8.(iii)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(f[x_] := Sin[x\^2]\/\(x + 1\)\), "\[IndentingNewLine]", \(\(Plot[f[x], {x, 0, \(3\/2\) \[Pi]}, PlotRange \[Rule] {\(-0.37\), 0.47}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] {True, False}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(\(f\_+\)[x_] := \(Abs[f[x]] + f[x]\)\/2\), "\[IndentingNewLine]", \(\(f\_-\)[x_] := \(Abs[f[x]] - f[x]\)\/2\), "\[IndentingNewLine]", \(\(Plot[\(f\_+\)[x], {x, 0, \(3\/2\) \[Pi]}, PlotRange \[Rule] {\(-0.03\), 0.47}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] {True, False}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(\(Plot[\(f\_-\)[x], {x, 0, \(3\/2\) \[Pi]}, PlotRange \[Rule] {\(-0.03\), 0.47}, AspectRatio \[Rule] Automatic, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] {True, False}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 6.4.4", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[Integrate::idiv]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(f[x_, y_] := \(x - y\)\/\((x + y)\)\^3\), "\[IndentingNewLine]", \(\(Plot3D[f[x, y], {x, 0.001, 1}, {y, 0.001, 1}, ColorFunction \[Rule] Hue, Background \[Rule] lightblue, ClipFill \[Rule] None, PlotPoints \[Rule] {80, 80}, ViewPoint \[Rule] {\(-8\), \(-2.4\), 1.4}, Boxed \[Rule] False, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\), "\[IndentingNewLine]", \(\(Plot3D[f[x, y], {x, 0.001, 1}, {y, 0.001, 1}, ColorFunction \[Rule] Hue, Background \[Rule] lightblue, ClipFill \[Rule] None, PlotRange \[Rule] All, PlotPoints \[Rule] {80, 80}, ViewPoint \[Rule] {\(-8\), \(-2.4\), 1.4}, Boxed \[Rule] False, Axes \[Rule] None, ImageSize \[Rule] 1000];\)\), "\[IndentingNewLine]", \(\[Integral]\_0\%1\((\[Integral]\_0\%1\( x\/\((x + y)\)\^3\) \[DifferentialD]y)\) \[DifferentialD]x\ \[IndentingNewLine]\), "\[IndentingNewLine]", \(Integrate[f[x, y], {y, 0, 1}, Assumptions -> {0 < x < 1}]\), "\[IndentingNewLine]", \(Integrate[%, {x, 0, 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 6.5.1 Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Simplify[ Integrate[ x\_1^2 + x\_2^2 + x\_3^2, {x\_3, 0, a - x\_1 - x\_2}] \[Equal] \((x\_1^2 + x\_2^2)\) \((a - x\_1 - x\_2)\) + \((a - x\_1 - x\_2)\)^3/ 3]\), "\[IndentingNewLine]", \(Simplify[ Integrate[ Integrate[ x\_1^2 + x\_2^2 + x\_3^2, {x\_3, 0, a - x\_1 - x\_2}], {x\_2, 0, a - x\_1}] \[Equal] x\_1^2 \((a - x\_1)\)^2/2 + \((a - x\_1)\)^4/ 6]\), "\[IndentingNewLine]", \(Integrate[ Integrate[ Integrate[ x\_1^2 + x\_2^2 + x\_3^2, {x\_3, 0, a - x\_1 - x\_2}], {x\_2, 0, a - x\_1}], {x\_1, 0, a}]\), "\[IndentingNewLine]", \(\(a = 3;\)\), "\[IndentingNewLine]", \(Integrate[ Boole[x\_1 + x\_2 + x\_3 \[LessEqual] a] \((x\_1^2 + x\_2^2 + x\_3^2)\), {x\_1, 0, a}, {x\_2, 0, a}, {x\_3, 0, a}]\), "\[IndentingNewLine]", \(\(a = 7;\)\), "\[IndentingNewLine]", \(\(\(Integrate[ Boole[x\_1 + x\_2 + x\_3 \[LessEqual] a] \((x\_1^2 + x\_2^2 + x\_3^2)\), {x\_1, 0, a}, {x\_2, 0, a}, {x\_3, 0, a}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(a = 11.5;\)\), "\n", \(\(b1 = 0.35;\)\), "\n", \(\(b2 = 0.45;\)\), "\n", \(\(x1 = 3;\)\), "\n", \(\(x2 = 4.5;\)\), "\n", \(\(Graphics3D[{Text["\", {x1, 0, 0}], Text["\", {0, x2, 0}], Text["\", {0, 0, a - x1 - x2}], Text["\", {a, 0, 0}], Text["\", {0, a - x1, 0}], {RGBColor[1, 0, 0], Line[{{a, 0, 0}, {0, a, 0}, {0, 0, a}, {b1\ a, 0, \((1 - b1)\)\ a}}], Line[{{b2\ a, 0, \((1 - b2)\)\ a}, {a, 0, 0}}], Line[{{x1, 0, 0}, {x1, a - x1, 0}, {0, a - x1, 0}}]}, {RGBColor[ 1, 0, 0], Line[{{x1, x2, 0}, {0, x2, 0}}], Line[{{x1, x2, 0}, {x1, x2, a - x1 - x2}, {0, 0, a - x1 - x2}}], Line[{{0, 0, 0}, {a\ x1/\((x1 + x2)\), a\ x2/\((x1 + x2)\), 0}}]}, {RGBColor[1, 0, 0], Line[{{a\ x1/\((x1 + x2)\), a\ x2/\((x1 + x2)\), 0}, {0, 0, a}}]}, {RGBColor[0, 0.5, 0], Line[{{a + 2, 0, 0}, {0, 0, 0}, {0, a + 2, 0}}], Line[{{0, 0, 0}, {0, 0, a + 2}}]}, Point[{x1, 0, 0}], Point[{0, x2, 0}], Point[{0, 0, a - x1 - x2}]}, {ViewPoint \[Rule] {18, 9, 13}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None}];\)\), "\n", \(\(Show[%, Background \[Rule] lightblue, ImageSize \[Rule] 600, DefaultFont \[Rule] 12. ];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 6.5.2 (Intersection of two cylinders) New Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ Boole[x1^2 + x2^2 \[LessEqual] 1 && x1^2 + x3^2 \[LessEqual] 1], {x1, \(-1\), 1}, {x2, \(-1\), 1}, {x3, \(-1\), 1}]\), "\[IndentingNewLine]", \(Print["\"]\), "\n", \(b[x_] := Sqrt[1 - x^2]\), "\n", \(Integrate[ Integrate[ Integrate[1, {x3, \(-b[x1]\), b[x1]}], {x2, \(-b[x1]\), b[x1]}], {x1, \(-1\), 1}]\), "\n", \(Print["\"]\), "\n", \(f[\(x\_1\) _] = Integrate[Sqrt[1 - x\_1^2], x\_\(\(1\)\(\ \)\)]\), "\n", \(f[b[x\_1]]\), "\n", \(Integrate[x2\ Sqrt[1 - x2^2], {x2, 0, 1}]\), "\n", \(\(\(Integrate[ArcCos[x2], {x2, 0, 1}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(c1 = ParametricPlot3D[ Append[{Cos[s], Sin[s], t}, SurfaceColor[RGBColor[0.7, 1, 1]]] // Evaluate, {s, \(-\[Pi]\), \[Pi]}, {t, \(-2\), 2}, PlotPoints \[Rule] {121, 121}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(c2 = ParametricPlot3D[{Cos[s], t, Sin[s]} // Evaluate, {s, \(-\[Pi]\), \[Pi]}, {t, \(-2\), 2}, PlotPoints \[Rule] {121, 121}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(l1 = ParametricPlot3D[{Cos[s], Sin[s], Sin[s]}, {s, \(-\[Pi]\), \[Pi]}, PlotPoints \[Rule] 301, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(l2 = ParametricPlot3D[{Cos[s], Sin[s], \(-Sin[s]\)}, {s, \(-\[Pi]\), \[Pi]}, PlotPoints \[Rule] 301, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(Show[c1, c2, l1, l2, ViewPoint \[Rule] {5, 0.5, 2}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, ImageSize \[Rule] 900, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(d1 = ParametricPlot3D[ Append[Boole[\(-Sin[s]\) \[LessEqual] t \[LessEqual] Sin[s]] {Cos[ s], Sin[s], t}, SurfaceColor[RGBColor[0.7, 1, 1]]] // Evaluate, {s, \(-\[Pi]\), \[Pi]}, {t, \(-1\), 1}, PlotPoints \[Rule] {301, 301}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(d2 = ParametricPlot3D[ Append[Boole[ Sin[s] \[LessEqual] t \[LessEqual] \(-Sin[s]\)] {Cos[s], Sin[s], t}, SurfaceColor[RGBColor[0.7, 1, 1]]] // Evaluate, {s, \(-\[Pi]\), \[Pi]}, {t, \(-1\), 1}, PlotPoints \[Rule] {301, 301}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(d3 = ParametricPlot3D[ Boole[\(-Sin[s]\) \[LessEqual] t \[LessEqual] Sin[s]] {Cos[s], t, Sin[s]} // Evaluate, {s, \(-\[Pi]\), \[Pi]}, {t, \(-1\), 1}, PlotPoints \[Rule] {301, 301}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(d4 = ParametricPlot3D[ Boole[Sin[s] \[LessEqual] t \[LessEqual] \(-Sin[s]\)] {Cos[s], t, Sin[s]} // Evaluate, {s, \(-\[Pi]\), \[Pi]}, {t, \(-1\), 1}, PlotPoints \[Rule] {301, 301}, LightSources \[Rule] {{{1, 0, 1}, RGBColor[1, 0, 0]}, {{1, 1, 1}, RGBColor[0, 1, 0]}, {{0, 1, 1}, RGBColor[0, 0, 1]}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(Show[d1, d2, d3, d4, l1, l2, ViewPoint \[Rule] {5, 0.5, 2}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, ImageSize \[Rule] 900, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(e1 = ParametricPlot3D[ Append[{Cos[s], Sin[s], t}, FaceForm[RGBColor[0.9, 1, 0.5], darkblue]] // Evaluate, {s, \(-\[Pi]\), \[Pi]}, {t, \(-2\), 2}, PlotPoints \[Rule] {121, 121}, Lighting \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(e2 = ParametricPlot3D[ Append[{Cos[s], t, Sin[s]}, FaceForm[darkblue, RGBColor[0.9, 1, 0.5]]] // Evaluate, {s, \(-\[Pi]\), \[Pi]}, {t, \(-2\), 2}, PlotPoints \[Rule] {121, 121}, Lighting \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[e1, e2, l1, l2, ViewPoint \[Rule] {2.5, 2, 1}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, ImageSize \[Rule] 900, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Example 6.5.3 (Sharpening a pencil) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Tan[\[Pi]/6]\), "\[IndentingNewLine]", \(Tan[\[Pi]/12]\), "\[IndentingNewLine]", \(Tan[\[Pi]/2 - \[Pi]/12]\), "\[IndentingNewLine]", \(\(a = 3.5;\)\), "\[IndentingNewLine]", \(6 Integrate[ Integrate[ Integrate[ 1, {x\_3, \(-\((2 + Sqrt[3])\)\) Sqrt[x\_1^2 + x\_2^2], 0}], {x\_2, \(-Sqrt[3]\) x\_1/3, Sqrt[3] x\_1/3}], {x\_1, 0, a}]\), "\[IndentingNewLine]", \(Simplify[% == 12 \((2 + Sqrt[3])\)/36\ \((4 + 3 Log[3])\) a^3]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 6.6.5 (Volume of truncated cone) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Simplify[ Integrate[\((1 - s)\)^2, {s, 0, h/t}] \[Equal] \((1 - \((1 - h/t)\)^3)\)/ 3]\), "\[IndentingNewLine]", \(\(\(Simplify[ t\ a^2 \((1 - \((1 - h/t)\)^3)\)/3 /. t \[Rule] a\ h/\((a - b)\)]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(t = 2;\)\), "\[IndentingNewLine]", \(\(smax = 0.4;\)\), "\[IndentingNewLine]", \(\[Psi][\[Alpha]_, s_] := \((1 - s)\) {5 + Cos[\[Alpha]] + 0.08 Cos[15 \[Alpha]], Sin[\[Alpha]] + 0.08 Cos[15 \[Alpha]], 0} + {0, 0, s\ t}\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot3D[ Append[\[Psi][\[Alpha], s], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {s, 0, smax}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, ViewPoint \[Rule] {2, 1, 0.9}, PlotPoints \[Rule] {201, 101}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 900, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\[Chi][\[Alpha]_, r_] := \((1 - smax)\) {5 + r \((Cos[\[Alpha]] + 0.08 Cos[15 \[Alpha]])\), r \((Sin[\[Alpha]] + 0.08 Cos[15 \[Alpha]])\), 0} + {0, 0, smax\ t}\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot3D[ Append[\[Chi][\[Alpha], r], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {r, 0, 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, ViewPoint \[Rule] {2, 1, 0.9}, PlotPoints \[Rule] {31, 21}, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 900, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[p1, p2, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 6.6.7 (Newton's potential of a ball) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[ r/Sqrt[r^2 + \((y\_3 - a)\)^2], {r, 0, Sqrt[R^2 - y\_3^2]}, Assumptions \[Rule] {y\_3^2 \[LessEqual] R^2, a - y\_3 > 0}]\), "\[IndentingNewLine]", RowBox[{\(Integrate[1/2\ %, {y\_3, \(-R\), R}, Assumptions \[Rule] {a > R > 0}]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ "Print", "[", "\"\\"Italic\"]\)\_3\),r) in order of \ integration\>\"", "]"}], "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(thick = 0.004;\), "\[IndentingNewLine]", \(R = 2;\), "\[IndentingNewLine]", \(a = Graphics3D[{Thickness[thick], {RGBColor[0, 0.5, 0], Line[{{0, 0, 0}, {\[Pi] + 0.5, 0, 0}}]}}];\), "\n", \(y3 = Graphics3D[{Thickness[thick], {RGBColor[0, 0.5, 0], Line[{{0, 0, 0}, {0, 2.5, 0}}]}}];\), "\n", \(r = Graphics3D[{Thickness[thick], {RGBColor[0, 0.5, 0], Line[{{0, 0, 0}, {0, 0, 2.5}}]}}];\), "\[IndentingNewLine]", \(t1 = Graphics3D[ Text["\<\[Alpha]-axis\>", {\[Pi] + 0.8, 0, 0}]];\), "\[IndentingNewLine]", RowBox[{ RowBox[{"t2", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\)\_3\)-axis\>\"", ",", \({0, 2.9, 0}\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"t3", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\* StyleBox[\"r\",\nFontSlant->\"Italic\"]\)-axis\>\"", ",", \({0, 0, 2.7}\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", \(p1 = ParametricPlot3D[ Append[{\[Alpha], y3, Sqrt[R^2 - y3^2]}, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {y3, \(-R\), R}, PlotPoints \[Rule] {100, 51}, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(p2 = ParametricPlot3D[ Append[{\[Pi], s\ y3, s\ Sqrt[R^2 - y3^2]}, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {y3, \(-R\), R}, {s, 0, 1}, PlotPoints \[Rule] {50, 15}, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(Show[ a, y3, r, t1, t2, t3, p1, p2, ViewPoint \[Rule] {2, 2, 0.5}, \[IndentingNewLine]LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, \[IndentingNewLine]Background \[Rule] lightblue, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, ImageSize \[Rule] 900, TextStyle \[Rule] {FontFamily -> "\", FontSize \[Rule] 14. }, DisplayFunction \[Rule] $DisplayFunction];\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Formula (6.35) Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-4\)\ h} + {x, y}, {x, y}, R[a] . {2\ b, \(-4\)\ h} + {x, y}, R[a] . {0, \(-3\)\ h} + {x, y}}]\), "\[IndentingNewLine]", \(\(Plot[{7.5 + 0.1\ x^2, 1.5 + 1/24 - 1/24\ x^2}, {x, 1, 5}, AspectRatio \[Rule] 1, PlotStyle \[Rule] {AbsoluteThickness[1]}, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Graphics[{{AbsoluteThickness[1], Line[{{\(-11.5\), \(-1\)}, {\(-11.5\), 9}}], Line[{{\(-12\), 0}, {\(-4\), 0}}], Line[{{\(-1\), 0}, {7, 0}}], Line[{{\(-0.5\), \(-1\)}, {\(-0.5\), 9}}], Line[{{1, 1.5}, {1, 7.6}}], Line[{{5, 0.5}, {5, 10}}]}, {AbsoluteThickness[2], Line[{{2.5, 1.5 - 1/24\ 2.5\ 2.5}, {2.5, 7.5 + 0.1\ 2.5\ 2.5}}]}, {Dashing[{0.01, 0.01}], AbsoluteThickness[0.5], Line[{{2.5, 0}, {2.5, 1.5 - 1/24\ 2.5\ 2.5}}], Line[{{\(-0.5\), 6}, {2.5, 6}}]}}];\)\), "\n", \(\(Graphics[{{AbsoluteThickness[1], Line[{{\(-10\), 2}, {\(-10\), 6}}], Line[{{\(-10\), 6}, {\(-6\), 6}}], Line[{{\(-6\), 6}, {\(-6\), 2}}], Line[{{\(-6\), 2}, {\(-10\), 2}}]}, {AbsoluteThickness[2], Line[{{\(-8.5\), 2}, {\(-8.5\), 6}}]}, {Dashing[{0.01, 0.01}], AbsoluteThickness[0.5], Line[{{\(-8.5\), 0}, {\(-8.5\), 2}}], Line[{{\(-11.5\), 5}, {\(-8.5\), 5}}]}}, AspectRatio \[Rule] Automatic, Axes \[Rule] True, Ticks \[Rule] None];\)\), "\n", \(\(Graphics[{{AbsoluteThickness[1], Line[{{\(-5\), 3}, {\(-1.5\), 3}}]}, arrow[\(-1.5\), 3, 0.15, 0.1, \[Pi]/2], arrow[\(-8.5\), 4, 0.15, 0.1, 0], arrow[2.5, 4.5, 0.15, 0.1, 0], Text["\", {\(-12\), 5}], Text["\<(y1...yn-1)\>", {\(-8.5\), \(-0.5\)}], Text["\<(x1...xn-1)\>", {2.5, \(-0.5\)}], Text["\", {\(-3\), 3}], Text["\", {\(-2\), 6}], Text["\", {\(-12\), 8}], Text["\", {\(-1\), 8}], Text["\", {\(-4.5\), \(-0.5\)}], Text["\", {6, \(-0.5\)}]}];\)\), "\n", \(\(Show[%%%%, %%%, %%, %, DefaultColor \[Rule] darkblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{\(-13\), 7}, {\(-4.5\), 10.5}}, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction, DefaultFont \[Rule] 12. ];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Theorem 6.7.4 Illustration for the proof\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(g[x_] := \[ExponentialE]^\((\(-1\)/x)\);\)\), "\n", \(\(g1 = Plot[g[x], {x, 0.01, 1}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] All, PlotPoints \[Rule] 2000, \ Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(g21[x_] := g[x + 2]*g[\(-1\) - x];\)\), "\n", \(\(g2 = Plot[g21[x], {x, \(-1.99\), \(-1.01\)}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], AspectRatio \[Rule] 1/3, \ PlotPoints \[Rule] 2000, \ Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(pg21[x_] := NIntegrate[g21[t], {t, \(-1.99\), x}];\)\), "\n", \(\(h21[x_] := pg21[x]/pg21[\(-1\)];\)\), "\n", \(\(p1 = Plot[h21[x], {x, \(-1.99\), \(-1.01\)}, PlotStyle -> RGBColor[1, 0, 0], \ AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotPoints \[Rule] 2001, \ Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = Plot[h21[\(-x\)], {x, 1.01, 1.99}, \ PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotPoints \[Rule] 2001, \ Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = Plot[1, {x, \(-1\), 1}, \ PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotPoints \[Rule] 2001, \ Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p4 = Graphics[{Dashing[{0.01, 0.01}], {darkblue, Line[{{1, 0}, {1, 1}}]}}];\)\), "\n", \(\(p5 = Graphics[{Dashing[{0.01, 0.01}], {darkblue, Line[{{\(-1\), 0}, {\(-1\), 1}}]}}];\)\), "\[IndentingNewLine]", \(\(g4 = Show[p1, p2, p3, \ p4, p5, \ AspectRatio \[Rule] 1/2, \ AxesOrigin \[Rule] {\(-2\), 0}, \ Ticks \[Rule] None];\)\), "\n", \(\(Show[GraphicsArray[{{g1, p1}, {g2, g4}}], ImageSize \[Rule] 900, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\)}], "Input", FontColor->RGBColor[0, 0, 0.500008], Background->RGBColor[0.832044, 0.996109, 0.832044]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Theorem 6.7.4 New Illustration \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[StyleBox["An example of a partition of unity subordinate to an \ open covering. ", FontColor->RGBColor[0.0156252, 0.191409, 0.55079]]], "Text", Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(Off[General::spell];\)\), "\[IndentingNewLine]", \(\(Off[General::spell1];\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\(s[0]\)[x_] := 1/6*x^3;\)\ \), "\[IndentingNewLine]", \(\(\(s[1]\)[x_] := 1/6*\((\((1 + x)\)^3 - 4 x^3)\);\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \(s[x_] := 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ /; x < \(-2\);\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \(s[ x_] := \(s[0]\)[x + 2]\ \ /; \(-2\) < x < \(-1\);\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \(s[ x_] := \(s[1]\)[x + 1]\ \ /; \(-1\) < x < 0;\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \(s[x_] := \(s[1]\)[\(-x\) + 1] /; 0 < x < 1;\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \(s[x_] := \(s[0]\)[\(-x\) + 2] /; 1 < x < 2;\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \(s[x_] := 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ /; 2 < x;\)\), "\[IndentingNewLine]", \(\ \ \(spl[x_] := s[2 x];\)\), "\[IndentingNewLine]", \(\(spl0 = Plot[spl[x + 1], \ \ \ \ \ \ \ {x, \(-2\), 0}, \ \ \ \ \ \ \ \ \ PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] {0, 0.7}, AspectRatio \[Rule] Automatic, Ticks \[Rule] {Table[\(-2\) + 0.5\ k, {k, 0, 4}], {0, 0.5, 1}}, Background \[Rule] lightblue];\)\), "\[IndentingNewLine]", \(\(spl1 = Plot[spl[x + 1/2], \ {x, \(-1.5\), 0.5}, PlotStyle -> RGBColor[0, 1, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] {0, 0.7}, AspectRatio \[Rule] Automatic, Ticks \[Rule] {Table[\(-1.5\) + 0.5\ k, {k, 0, 4}], {0, 0.5, 1}}, Background \[Rule] lightblue];\)\), "\[IndentingNewLine]", \(\(spl2 = Plot[spl[x], \ \ \ \ \ \ \ \ \ \ \ \ \ {x, \(-1\), 1}, \ \ \ \ \ \ \ \ \ \ PlotStyle -> RGBColor[0, 0, 1], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] {0, 0.7}, AspectRatio \[Rule] Automatic, Ticks \[Rule] {Table[\(-1\) + 0.5\ k, {k, 0, 4}], {0, 0.5, 1}}, Background \[Rule] lightblue];\)\), "\[IndentingNewLine]", \(\(spl3 = Plot[spl[x - 1/2], {x, \(-0.5\), 1.5}, \ PlotStyle -> RGBColor[0.8, 0, 1], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] {0, 0.7}, AspectRatio \[Rule] Automatic, Ticks \[Rule] {Table[\(-0.5\) + 0.5\ k, {k, 0, 4}], {0, 0.5, 1}}, Background \[Rule] lightblue];\)\), "\[IndentingNewLine]", \(\(spl4 = Plot[spl[x - 1], \ \ \ \ \ \ {x, 0, 2}, \ \ \ \ \ \ \ \ \ \ \ \ \ PlotStyle -> RGBColor[0, 0.5, 0.5], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] {0, 0.7}, AspectRatio \[Rule] Automatic, Ticks \[Rule] {Table[\(-2\) + 0.5\ k, {k, 0, 4}], {0, 0.5, 1}}, Background \[Rule] lightblue];\)\), "\[IndentingNewLine]", \(\(Show[spl0, spl1, spl2, spl3, spl4, PlotRange \[Rule] {0, 0.7}, Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, Ticks \[Rule] {Table[\(-2\) + 0.5\ k, {k, 0, 8}], {0, 0.5, 1}}, ImageSize \[Rule] 800];\)\), "\[IndentingNewLine]", \(\(ssp = Plot[Sum[spl[x + j/2], {j, \(-2\), 2}], {x, \(-2\), 2}, PlotStyle -> RGBColor[0, 0, 1], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] {0, 1}, AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, Ticks \[Rule] {Table[\(-2\) + 0.5\ k, {k, 0, 8}], {0, 0.5, 1}}, ImageSize \[Rule] 800];\)\)}], "Input", FontColor->RGBColor[0, 0, 0.500008], Background->RGBColor[0.832044, 0.996109, 0.832044]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Example 6.10.2 (", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\[DoubleStruckCapitalR]\)]], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(-x\^2\)\)]], StyleBox["dx", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\@\[Pi]\)]], ")" }], "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\( \[ExponentialE]\^\(-x\^2\)\ \) \[DifferentialD]x\), "\[IndentingNewLine]", \(Integrate[ r\ \[ExponentialE]^\((\(-r^2\))\), {\[Alpha], \(-\[Pi]\), \[Pi]}, {r, 0, R}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 6.12.5 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(f[y_, z_] := \[ExponentialE]^\((\(-y^4\) z^2 - z^2)\)\), "\[IndentingNewLine]", \(Integrate[f[y, z], {z, \(-\[Infinity]\), \[Infinity]}, Assumptions \[Rule] {Im[y] == 0}]\), "\[IndentingNewLine]", \(Integrate[%, {y, \(-\[Infinity]\), \[Infinity]}]\), \ "\[IndentingNewLine]", \(Integrate[ f[y, 0], {y, \(-\[Infinity]\), \[Infinity]}]\), "\[IndentingNewLine]", \(Integrate[f[y, z], {y, \(-\[Infinity]\), \[Infinity]}, Assumptions \[Rule] {Im[z] \[Equal] 0, z \[NotEqual] 0}]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 7: THEORY", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Section 7.3. (Area equals volume divided by length)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"];\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\[IndentingNewLine]", \(<< LinearAlgebra`Orthogonalization`\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\)\_1\)\[Phi](\!\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\)), \!\(\(\* StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\_2\)\[Phi](\!\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\)) and \!\(\[Nu]\_1\)(\!\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(a[1] = {1, 2, 0};\), "\[IndentingNewLine]", \(a[0] = 0 a[1];\), "\[IndentingNewLine]", \(a[2] = {0, 1, 0};\), "\[IndentingNewLine]", \(a[3] = a[1] + a[2];\), "\[IndentingNewLine]", \(a[4] = Normalize[Cross[a[1], a[2]]];\), "\[IndentingNewLine]", \(Do[ a[j_] := a[j - 4] + a[4], {j, 5, 7, 1}];\), "\[IndentingNewLine]", \(ta = Table[Point[a[j]], {j, 0, 7}];\), "\[IndentingNewLine]", \(p = Graphics3D[{AbsolutePointSize[7], darkblue, ta}, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(l[ 1] = Line[{a[0], a[1], a[3], a[2], a[0]}];\), "\[IndentingNewLine]", \(l[2] = Line[{a[4], a[5], a[7], a[6], a[4]}];\), "\[IndentingNewLine]", \(Do[ l[j_] := Line[{a[j - 3], a[j + 1]}], {j, 3, 6, 1}];\), "\[IndentingNewLine]", \(l = Graphics3D[{RGBColor[1, 0, 0], Table[l[j], {j, 1, 6}]}, ViewPoint \[Rule] 100 {1, 2, 1}, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"t", "=", RowBox[{"Graphics3D", "[", RowBox[{ RowBox[{"{", RowBox[{\(Text["\<0\>", a[0] + {0.05, \(-0.1\), 0}]\), ",", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\(\* StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\_1\)\[Phi](\!\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\))\>\"", ",", \(a[1] + {0.2, 0, 0}\)}], "]"}], ",", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\(\* StyleBox[\"D\",\nFontSlant->\"Italic\"]\)\_2\)\[Phi](\!\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\))\>\"", ",", \(a[2] + {0.2, \(-0.1\), 0.05}\)}], "]"}], ",", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\[Nu]\_1\)(\!\(\* StyleBox[\"y\",\nFontSlant->\"Italic\"]\))\>\"", ",", \(a[4] + {0.4, \(-0.4\), 0.35}\)}], "]"}]}], "}"}], ",", \(DisplayFunction \[Rule] Identity\)}], "]"}]}], ";"}], " "}], "\[IndentingNewLine]", \(Show[p, l, t, TextStyle \[Rule] {FontFamily -> "\", FontSize \[Rule] 14. }, Boxed \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction \[Rule] $DisplayFunction];\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.4.1 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["'s definition of the complete elliptic integral of the second \ kind with modulus ", FontVariations->{"CompatibilityType"->0}], StyleBox["k ", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["is not the usual one, as customary in pure mathematics. In fact, \ ", FontVariations->{"CompatibilityType"->0}], StyleBox["E", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["(", FontVariations->{"CompatibilityType"->0}], StyleBox["k", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[") in the book is given by EllipticE[k^2] in ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica.", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" Furthermore, it is unable to establish a trivial identity for \ this integral. ", FontVariations->{"CompatibilityType"->0}] }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \ \(Print["\"]\), "\[IndentingNewLine]", \(\[Phi][t_] := {\ Cos[t], Sin[t]}\), "\[IndentingNewLine]", \(Sqrt[ Sum[\((\[PartialD]\_t\ \(\[Phi][t]\)[\([j]\)])\)^2, {j, 1, 2}]]\), "\[IndentingNewLine]", RowBox[{\(FullSimplify[Integrate[%, {t, 0, x}]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\)emi-minor axis \!\(\* StyleBox[\"b\",\nFontSlant->\"Italic\"]\), semi-focal separation \!\(\* StyleBox[\"c\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)= \!\(\@\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\^2 - \* StyleBox[\"b\",\nFontSlant->\"Italic\"]\^2\)\)\>\"", "]"}], "\[IndentingNewLine]", \(\[Phi][t_] := {a\ Cos[t], b\ Sin[t]}\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\[IndentingNewLine]", \(a = 2;\), "\[IndentingNewLine]", \(b = 1;\), "\[IndentingNewLine]", \(c = Sqrt[a^2 - b^2];\), "\[IndentingNewLine]", \(p1 = ParametricPlot[\[Phi][t], {t, \(-\[Pi]\), \[Pi]}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-1.15\), 1.15}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Axes \[Rule] True, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(p2 = Graphics[{AbsolutePointSize[5], darkblue, Point[{c, 0}], Point[{\(-c\), 0}], Point[\[Phi][\[Pi]/5]], Point[\[Phi][\[Pi]/2]]}, PlotRange \[Rule] {0, 3}, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(p3 = Graphics[{darkblue, Line[{{c, 0}, \[Phi][\[Pi]/5], {\(-c\), 0}}], Line[{{c, 0}, \[Phi][\[Pi]/2], {\(-c\), 0}}]}, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"te", "=", RowBox[{"Graphics", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Text", "[", RowBox[{"\"\<\!\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\>\"", ",", \({0.7, 0.7}\)}], "]"}], ",", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\* StyleBox[\"b\",\nFontSlant->\"Italic\"]\)\>\"", ",", \({\(-0.1\), 0.5}\)}], "]"}], ",", RowBox[{"Text", "[", RowBox[{"\"\<\!\(\* StyleBox[\"c\",\nFontSlant->\"Italic\"]\)\>\"", ",", \({0.7, \(-0.1\)}\)}], "]"}]}], "}"}], ",", \(DisplayFunction \[Rule] Identity\)}], "]"}]}], ";"}], " "}], "\[IndentingNewLine]", \(Show[p1, p2, p3, te, TextStyle \[Rule] {FontFamily -> "\", FontSize \[Rule] 14. }, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)= 2\!\(\* StyleBox[\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\\\ \)]\)and \!\(\* StyleBox[\"b\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)= 1\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)the eccentricity \!\(\* StyleBox[\"e\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)equals\>\"", "]"}], "\[IndentingNewLine]", \(e = c/a\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(N[ 4 a\ EllipticE[e^2]]\), "\[IndentingNewLine]", RowBox[{\(Clear[a, b, c, e]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Sqrt[ Sum[\((\[PartialD]\_t\ \(\[Phi][t]\)[\([j]\)])\)^2, {j, 1, 2}]]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[%, {t, \(-\[Pi]\), \[Pi]}, Assumptions \[Rule] {a > b > 0}]]\), "\[IndentingNewLine]", RowBox[{\(FullSimplify[ Integrate[%, {t, \(-\[Pi]\), \[Pi]}, Assumptions \[Rule] {a > b > 0}] \[Equal] 4 a\ EllipticE[\((a^2 - b^2)\)/a^2]] /. {\((1 - b^2/a^2)\) -> e^2}\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\ \"]\), "\[IndentingNewLine]", \(\[Phi][t_] := {\ t - Sin[t], 1 - Cos[t]}\), "\[IndentingNewLine]", \(ParametricPlot[\[Phi][ t], {t, \(-1.5\), 2\ \[Pi] + 1.5}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-0.1\), 2.2}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Axes \[Rule] True, Ticks \[Rule] None, ImageSize \[Rule] 500];\), "\[IndentingNewLine]", \(Sqrt[ Sum[\((\[PartialD]\_t\ \(\[Phi][t]\)[\([j]\)])\)^2, {j, 1, 2}]]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[%, {t, 0, 2 \[Pi]}]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Examples 7.4.2 and 7.4.3 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(f[t_] := Sqrt[R^2 - t^2]\), "\[IndentingNewLine]", \(Simplify[1 + \(f'\)[t]^2]\), "\[IndentingNewLine]", \(\(\(2 Integrate[Sqrt[%], {t, \(-R\), R}, Assumptions \[Rule] {R > 0}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_] := f[\[Alpha]] {Cos[\[Alpha]], Sin[\[Alpha]]}\), "\[IndentingNewLine]", \(l[f_] := FullSimplify[ Sqrt[Sum[\((\[PartialD]\_\[Alpha]\ \ \(\[Phi][\[Alpha]]\)[\([j]\)])\)^2, {j, 1, 2}]], Im[\[Alpha]] \[Equal] 0]\), "\[IndentingNewLine]", \(l[f]\), "\[IndentingNewLine]", \(\(f[\[Alpha]] = R;\)\), "\[IndentingNewLine]", \(\(\(Simplify[Integrate[l[f], {\[Alpha], 0, 2 \[Pi]}], R > 0]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(<< Graphics`\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Clear[f]\), "\[IndentingNewLine]", \(f[a_] := 2 \((1 + Cos[\[Alpha]])\)\), "\[IndentingNewLine]", \(\(PolarPlot[f[\[Alpha]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Axes \[Rule] True, Ticks \[Rule] None, ImageSize \[Rule] 400];\)\), "\[IndentingNewLine]", \(Integrate[l[f], {\[Alpha], 0, 2 \[Pi]}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(r[a_] := Cos[2 \[Alpha]]\), "\[IndentingNewLine]", \(\(PolarPlot[r[\[Alpha]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Axes \[Rule] False, ImageSize \[Rule] 400];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(r[a_] := Cos[6 \[Alpha]]\), "\[IndentingNewLine]", \(\(PolarPlot[r[\[Alpha]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Axes \[Rule] False, ImageSize \[Rule] 400];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Section 7.4 II. Area", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Do[task = setdel[Format[\(\(Derivative[KroneckerDelta[j, 1], KroneckerDelta[j, 2]]\)[Subscript[f_, i_]]\)[k__]], TraditionalForm[\(D\_j\) f\_i[k]]]; task /. setdel \[Rule] SetDelayed, {j, 2}]\), "\[IndentingNewLine]", \(jacobimatrix[f_, x_List] := \ With[{y = \(Unique[y] &\) /@ x}, Table[D[\(f[y]\)[\([i]\)], y[\([j]\)]], {i, Length[f[y]]}, {j, Length[x]}] /. \ Thread[y \[Rule] x]]\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(d[j_]\)[f_, x_List] := \ With[{y = \(Unique[y] &\) /@ x}, Table[\[PartialD]\_\(y[\([j]\)]\)\ \(f[y]\)[\([i]\)], {i, Length[f[y]]}] /. \ Thread[y \[Rule] x]]\), "\[IndentingNewLine]", \(\(\(co[x_, n_] := Table[x\_j, {j, n}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(yv = co[y, 2];\)\), "\[IndentingNewLine]", \(\[Phi][{y1_, y2_}] := {\[Phi]\_1[y1, y2], \[Phi]\_2[y1, y2], \[Phi]\_3[ y1, y2]}\), "\[IndentingNewLine]", \(\[Phi][yv] // MatrixForm\), "\[IndentingNewLine]", \(\((jac = jacobimatrix[\[Phi], yv])\) // MatrixForm\), "\[IndentingNewLine]", \(Transpose[jac] . jac // MatrixForm\), "\[IndentingNewLine]", \(j1 = Sqrt[Det[Transpose[jac] . jac]]\), "\[IndentingNewLine]", \(\({d1, d2} = Table[\(d[j]\)[\[Phi], yv], {j, 2}];\)\), "\[IndentingNewLine]", \(d1 // MatrixForm\), "\[IndentingNewLine]", \(j2 = Sqrt[Sum[\(Cross[d1, d2]\)[\([j]\)]^2, {j, 3}]]\), "\[IndentingNewLine]", \(Simplify[j1 \[Equal] j2]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.4.5 (Torus) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := {\((2 + Cos[\[Theta]])\) Cos[\[Alpha]], \((2 + Cos[\[Theta]])\) Sin[\[Alpha]], Sin[\[Theta]]}\), "\[IndentingNewLine]", \(\(\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]];\)\), "\ \[IndentingNewLine]", \(\(\[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]];\)\), "\ \[IndentingNewLine]", \(c = Simplify[\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]]\[Cross]\ \[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]]]\), "\[IndentingNewLine]", \(j = Simplify[\@\(\[Sum]\+\(j = 1\)\%3 c\[LeftDoubleBracket]j\ \[RightDoubleBracket]\^2\), {\(-\(\[Pi]\/2\)\) < \[Theta] < \[Pi]\/2}]\), "\ \[IndentingNewLine]", \(Integrate[ j, {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \(-\[Pi]\), \[Pi]}]\)}], \ "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.4.6 (Cap of sphere and kissing number) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[N::meprec]\), "\[IndentingNewLine]", \(Off[Triangle::shdw]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := R {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]]}\), "\[IndentingNewLine]", \(\(R = 1;\)\), "\[IndentingNewLine]", \(\(\[Psi] = \[Pi]/8;\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\(ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 0]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \[Psi], \ \[Pi]/2}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, ViewPoint \[Rule] {2, 1, 0.9}, PlotPoints \[Rule] {71, 31}, Ticks \[Rule] False, Boxed \[Rule] False, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(Clear[R, \[Psi]]\), "\[IndentingNewLine]", \(\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]]\), "\ \[IndentingNewLine]", \(\[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]]\), "\ \[IndentingNewLine]", \(c = Simplify[\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]]\[Cross]\ \[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]]]\), "\[IndentingNewLine]", \(j = Simplify[\@\(\[Sum]\+\(j = 1\)\%3 c\[LeftDoubleBracket]j\ \[RightDoubleBracket]\^2\), {Im[R] \[Equal] 0, \(-\(\[Pi]\/2\)\) < \[Theta] < \[Pi]\/2}]\), "\ \[IndentingNewLine]", \(Integrate[ j, {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \[Psi], \[Pi]\/2}]\), "\ \[IndentingNewLine]", \(% /. \[Psi] \[Rule] 0\[IndentingNewLine]\), "\[IndentingNewLine]", \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(<< Geometry`Polytopes`\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(vert = Vertices[Hexagon];\)\), "\[IndentingNewLine]", \(u = Union[Table[ FullSimplify[ Norm[\ vert\[LeftDoubleBracket]1\[RightDoubleBracket] - vert\[LeftDoubleBracket]j\[RightDoubleBracket]]], {j, 1, 6}]]\), "\[IndentingNewLine]", \(\(p = Graphics[{PointSize[ .01], darkblue, Point /@ vert}, AspectRatio \[Rule] 1];\)\), "\[IndentingNewLine]", \(\(v = Join[vert, {vert[\([1]\)]}];\)\), "\[IndentingNewLine]", \(\(h = Graphics[{RGBColor[1, 0, 0], Line[v]}];\)\), "\[IndentingNewLine]", \(\(c = Join[vert, {{0, 0}}];\)\), "\[IndentingNewLine]", \(\(sphere[{x1_, x2_}]\)[\[Alpha]_] := {x1 + Cos[\[Alpha]]/2, x2 + Sin[\[Alpha]]/2}\), "\[IndentingNewLine]", \(\(s = ParametricPlot[ Evaluate[ Table[\(sphere[c[\([j]\)]]\)[\[Alpha]], {j, 1, 7}]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] darkblue, Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, Axes \[Rule] False, Ticks \[Rule] None, PlotPoints \[Rule] 99, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[p, h, s, ImageSize \[Rule] 800, Background \[Rule] lightblue, Axes \[Rule] False, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] $DisplayFunction];\)\ \[IndentingNewLine]\), \ "\[IndentingNewLine]", \(Print["\"]\), \ "\[IndentingNewLine]", \(\(sphere[{x1_, x2_, x3_}]\)[\[Alpha]_, \[Theta]_] := {x1 + Cos[\[Alpha]] Cos[\[Theta]], x2 + Sin[\[Alpha]] Cos[\[Theta]], x3 + Sin[\[Theta]]}\), "\[IndentingNewLine]", \(\(s1 = ParametricPlot3D[ Evaluate[ Append[\(sphere[{0, 0, 0}]\)[\[Alpha], \[Theta]], SurfaceColor[ RGBColor[0.7, 0.7, 0]]]], {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \[Pi]/ 3, \[Pi]/2}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, Axes \[Rule] False, Boxed \[Rule] False, Ticks \[Rule] None, PlotPoints \[Rule] 31, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(s2 = ParametricPlot3D[ Evaluate[ Append[\(sphere[{0, 0, 0}]\)[\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]]]], {\[Alpha], \(-3\) \[Pi]/ 2, 0}, {\[Theta], \(-\[Pi]\)/2, \[Pi]/3}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, Axes \[Rule] False, Boxed \[Rule] False, Ticks \[Rule] None, PlotPoints \[Rule] 79, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(s3 = ParametricPlot3D[ Evaluate[ Append[\(sphere[{0, 0, 2}]\)[\[Alpha], \[Theta]], SurfaceColor[ RGBColor[0.7, 0.7, 1]]]], {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \(-\[Pi]\)/ 2, 0}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, Axes \[Rule] False, Boxed \[Rule] False, Ticks \[Rule] None, PlotPoints \[Rule] 99, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(x[\[Alpha]_] := {Cos[\[Alpha]] Cos[\(-\[Pi]\)/6], Sin[\[Alpha]] Cos[\(-\[Pi]\)/6], 2 + Sin[\(-\[Pi]\)/6]};\)\), "\[IndentingNewLine]", \(\(step = \[Pi]/30;\)\), "\n", \(\(lines = Graphics3D[ Table[{RGBColor[1, 0, 0], Line[{1.35 x[\[Alpha]], {0, 0, 0}}]}, {\[Alpha], \(-\[Pi]\) + step, \[Pi], step}]];\)\), "\n", \(\(Show[s1, s2, s3, lines, AspectRatio \[Rule] 1, ViewPoint \[Rule] {4, 4, 1.4}, Background \[Rule] lightblue, Axes \[Rule] False, Boxed \[Rule] False, Ticks \[Rule] None, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\)\[IndentingNewLine]\), "\ \[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(Simplify[ 4/\((2 - Sqrt[3])\) \[Equal] 8 + 4 Sqrt[3]]\), "\[IndentingNewLine]", \(N[4/\((2 - Sqrt[3])\)]\), "\[IndentingNewLine]", \(N[Sqrt[8 - 8/Sqrt[5]]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(sphere[{x1_, x2_, x3_}]\)[\[Alpha]_, \[Theta]_] := {x1 + Cos[\[Alpha]] Cos[\[Theta]], x2 + Sin[\[Alpha]] Cos[\[Theta]], x3 + Sin[\[Theta]]}\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(<< Graphics`Polyhedra`\), "\[IndentingNewLine]", \(\(Show[Graphics3D[Icosahedron[]], ViewPoint \[Rule] {4, 2, 1}, DefaultColor -> RGBColor[0.7, 0.7, 1], LightSources \[Rule] {{{1, 0, 0}, RGBColor[0, 0, 1]}, {{0, 1, 0}, RGBColor[0, 0, 1]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, Boxed \[Rule] False, ImageSize \[Rule] 800];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(\(vert = Vertices[Icosahedron];\)\), "\[IndentingNewLine]", \(u = Union[Table[ FullSimplify[ Norm[\ vert\[LeftDoubleBracket]1\[RightDoubleBracket] - vert\[LeftDoubleBracket]j\[RightDoubleBracket]]], {j, 1, 12}]]\), "\[IndentingNewLine]", \(Table[N[u[\([j]\)]], {j, 1, 4}]\), "\[IndentingNewLine]", \(f = FullSimplify[ Sqrt[8 - 8/Sqrt[5]]/u[\([4]\)]]\ \), "\[IndentingNewLine]", \(\(v = Table[Simplify[ f\ vert\[LeftDoubleBracket]j\[RightDoubleBracket]], {j, 1, 12}];\)\), "\[IndentingNewLine]", \(\(c = Join[v, {{0, 0, 0}}];\)\), "\[IndentingNewLine]", \(\(s = ParametricPlot3D[ Evaluate[ Table[Append[\(sphere[c[\([j]\)]]\)[\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]]], {j, 1, 13}]], {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \(-\[Pi]\)/ 2, \[Pi]/2}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, Axes \[Rule] False, Boxed \[Rule] False, Ticks \[Rule] None, PlotPoints \[Rule] 99, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[s, ImageSize -> 1200, ViewPoint \[Rule] {4, 2, 1}, Background \[Rule] lightblue, Axes \[Rule] False, Boxed \[Rule] False, Ticks \[Rule] None, DisplayFunction \[Rule] $DisplayFunction];\)\ \)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.4.7 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(\[Phi][s_]\)[\[Alpha]_, \[Theta]_] := 1/2 {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]], s}\), "\[IndentingNewLine]", \(\[Phi]\[Alpha] = \[PartialD]\_\[Alpha]\ \(\[Phi][ 1]\)[\[Alpha], \[Theta]]\), "\[IndentingNewLine]", \(\[Phi]\[Theta] = \[PartialD]\_\[Theta]\ \(\[Phi][ 1]\)[\[Alpha], \[Theta]]\), "\[IndentingNewLine]", \(n1 = Simplify[Sqrt[ Sum[\[Phi]\[Alpha][\([j]\)]^2, {j, 1, 4}]], {\(-\[Pi]\)/ 2 < \[Theta] < \[Pi]/2}]\), "\[IndentingNewLine]", \(n2 = Simplify[Sqrt[ Sum[\[Phi]\[Theta][\([j]\)]^2, {j, 1, 4}]], {\(-\[Pi]\)/ 2 < \[Theta] < \[Pi]/2}]\), "\[IndentingNewLine]", \(\[Phi]\[Alpha] . \[Phi]\[Theta]\), "\[IndentingNewLine]", \(n1\ n2\), "\[IndentingNewLine]", \(2 Integrate[ n1\ n2, {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \(-\[Pi]\)/2, \[Pi]/ 2}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.4.8 (Newton's potential of a sphere)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(1\/\(4 \[Pi]\)\) Integrate[\(\(R\^2\) Cos[\[Theta]]\)\/\@\(R\^2 + a\^2 - 2 a\ R\ Sin[\ \[Theta]]\), {\[Theta], \(-\(\[Pi]\/2\)\), \[Pi]\/2}, {\[Alpha], \(-\[Pi]\), \ \[Pi]}, Assumptions -> {0 < a < R}]\), "\[IndentingNewLine]", \(\(1\/\(4 \[Pi]\)\) Integrate[\(\(R\^2\) Cos[\[Theta]]\)\/\@\(R\^2 + a\^2 - 2 a\ R\ Sin[\ \[Theta]]\), {\[Theta], \(-\(\[Pi]\/2\)\), \[Pi]\/2}, {\[Alpha], \(-\[Pi]\), \ \[Pi]}, Assumptions -> {0 < R < a}]\), "\[IndentingNewLine]", \(\(R = 1\ ;\)\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\), "\[IndentingNewLine]", \(\(p1 = Plot[\(-R\), {a, 0, 1}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = Plot[\(-\(R\^2\/a\)\), {a, 1, 6}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(b1 = Graphics[{lightgreen, Polygon[{{0, 0}, {0, \(-1\) - 0.05}, {1, \(-1\) - 0.05}, {1, 0}}]}];\)\), "\[IndentingNewLine]", \(\(Show[b1, p1, p2, PlotRange \[Rule] {\(-1\) - 0.05, 0}, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.4.10 (Area of two-sphere)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(h[y1_, y2_] := Sqrt[1 - \((y1^2 + y2^2)\)]\), "\[IndentingNewLine]", \({\[PartialD]\_\(y\_1\)h[y\_1, y\_2], \[PartialD]\_\(y\_2\)h[y\_1, y\_2]}\), "\[IndentingNewLine]", \(FullSimplify[ Sqrt[1 + \((\[PartialD]\_\(y\_1\)h[y\_1, y\_2])\)^2 + \((\[PartialD]\_\(y\_2\)h[y\_1, y\_2])\)^2], {Im[y\_1] \[Equal] Im[y\_2] \[Equal] 0}]\), "\[IndentingNewLine]", \(PowerExpand[ FullSimplify[ Sqrt[1 + \((\[PartialD]\_\(y\_1\)h[y\_1, y\_2])\)^2 + \((\[PartialD]\_\(y\_2\)h[y\_1, y\_2])\)^2], {Im[y\_1] \[Equal] Im[y\_2] \[Equal] 0}]]\), "\[IndentingNewLine]", \(2 Integrate[ Boole[y1^2 + y2^2 < 1]/h[y1, y2], {y1, \(-1\), 1}, {y2, \(-1\), 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.4.11 (Hyperarea of three-sphere)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(h[y1_, y2_, y3_] := Sqrt[1 - \((y1^2 + y2^2 + y3^2)\)]\), "\[IndentingNewLine]", \({\[PartialD]\_\(y\_1\)h[y\_1, y\_2, y\_3], \[PartialD]\_\(y\_2\)h[y\_1, y\_2, y\_3], \[PartialD]\_\(y\_3\)h[y\_1, y\_2, y\_3]}\), "\[IndentingNewLine]", \(FullSimplify[ Sqrt[1 + \((\[PartialD]\_\(y\_1\)h[y\_1, y\_2, y\_3])\)^2 + \((\[PartialD]\_\(y\_2\)h[y\_1, y\_2, y\_3])\)^2 + \((\[PartialD]\_\(y\_3\)h[y\_1, y\_2, y\_3])\)^2], {Im[y\_1] \[Equal] Im[y\_2] \[Equal] Im[y\_3] == 0}]\), "\[IndentingNewLine]", \(2 Integrate[ Boole[y1^2 + y2^2 + y3^2 < 1]/h[y1, y2, y3], {y1, \(-1\), 1}, {y2, \(-1\), 1}, {y3, \(-1\), 1}]\), "\[IndentingNewLine]", \(2 Integrate[ Cos[\[Theta]] r^2/Sqrt[ 1 - r^2], {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \(-\[Pi]\)/ 2, \[Pi]/2}, {r, 0, 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Formula (7.32)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Phi][y_] := 3/\((1 + y^3)\) {y^2, y}\), "\[IndentingNewLine]", \(\(v = {1, \(-1\)};\)\), "\[IndentingNewLine]", \(\[CapitalPsi][t_, y_] := \[Phi][y] + t\ v\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[ Evaluate[ Table[\[CapitalPsi][t, y], {y, 0, 1, 0.005}]], {t, \(-0.3\), 0.3}, PlotStyle \[Rule] darkblue, PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[Evaluate[\[Phi][y]], {y, 0, 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = ParametricPlot[Evaluate[\[Phi][y]] + {0.003, 0.003}, {y, 0, 1}, PlotStyle \[Rule] RGBColor[1, 0, 0], PlotPoints \[Rule] 200, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(\(Show[p1, p2, p3, AspectRatio \[Rule] Automatic, Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]]}\), "\[IndentingNewLine]", \(\(v = {1, \(-1\), 1};\)\), "\[IndentingNewLine]", \(\[CapitalPsi][t_, \[Alpha]_, \[Theta]_] := \[Phi][\[Alpha], \[Theta]] + t\ v\), "\[IndentingNewLine]", \(\(p1 = Table[Graphics3D[{darkblue, Line[{\[CapitalPsi][\(-0.5\), \[Alpha], \[Theta]], \ \[CapitalPsi][0.5, \[Alpha], \[Theta]]}]}, DisplayFunction \[Rule] Identity], {\[Alpha], 0, \[Pi]/3, \[Pi]/101}, {\[Theta], 0, \[Pi]/5, \[Pi]/101}];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[1, 0, 0]] // Evaluate], {\[Alpha], \(-\[Pi]\)/20, \[Pi]/ 2.5}, {\[Theta], \(-\[Pi]\)/20, \[Pi]/ 4.5}, \[IndentingNewLine]LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, \[IndentingNewLine]PlotPoints \[Rule] 20, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(\(Show[p1, p2, ViewPoint \[Rule] 10 {1.3, \(-2.4\), 2}, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]]}\), "\[IndentingNewLine]", \(\(v = {1, \(-1\), 1};\)\), "\[IndentingNewLine]", \(\[CapitalPsi][t_, \[Alpha]_, \[Theta]_] := \[Phi][\[Alpha], \[Theta]] + t\ v\), "\[IndentingNewLine]", \(\(p1 = Table[ParametricPlot3D[ Append[\[CapitalPsi][t, \[Alpha], \[Theta]], SurfaceColor[darkblue] // Evaluate], {\[Alpha], 0, \[Pi]/3}, {\[Theta], 0, \[Pi]/5}, PlotPoints \[Rule] 20, DisplayFunction \[Rule] Identity], {t, \(-0.5\), 0.5, 0.05}];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[1, 0, 0]] // Evaluate], {\[Alpha], \(-\[Pi]\)/20, \[Pi]/ 2.5}, {\[Theta], \(-\[Pi]\)/20, \[Pi]/ 4.5}, \[IndentingNewLine]PlotPoints \[Rule] 20, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[p1, p2, ViewPoint \[Rule] 10 {1.3, \(-2.4\), 2}, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.8.4 (Newton vector field)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ RowBox[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"list", " ", "of", " ", StyleBox["j", FontSlant->"Italic"]}], "-", RowBox[{ "th", " ", "partial", " ", "derivatives", " ", "of", " ", StyleBox["j", FontSlant->"Italic"]}], "-", \(th\ component\ function\ of\ vector\ field\)}], ",", " ", RowBox[{\(for\ 1\), " ", "\[LessEqual]", " ", StyleBox["j", FontSlant->"Italic"], " ", "\[LessEqual]", " ", StyleBox["n", FontSlant->"Italic"]}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{\(\(p[f_, x_List] := \ With[{y = \(Unique[y] &\) /@ x}, Table[D[\(f[y]\)[\([j]\)], y[\([j]\)]], {j, Length[f[y]]}] /. \ Thread[y \[Rule] x]]\)\(\[IndentingNewLine]\) \( (*\ divergence\ of\ vector\ field\ on\ \[DoubleStruckCapitalR]\^n\ *) \ \)\), " "}], "\[IndentingNewLine]", RowBox[{\(\(d[f_, x_List] := Simplify[ Sum[\(p[f, x]\)[\([j]\)], {j, 1, Length[x]}]]\)\(\[IndentingNewLine]\) \( (*\ gradient\ of\ function\ on\ \[DoubleStruckCapitalR]\^n\ *) \)\), " "}], "\[IndentingNewLine]", RowBox[{\(\(gr[g_, x_List] := With[{y = \(Unique[y] &\) /@ x}, Table[D[g[y], y[\([j]\)]], {j, Length[y]}] /. \ Thread[y \[Rule] x]]\)\(\[IndentingNewLine]\) \( (*\ Newton\ vector\ field\ on\ \[DoubleStruckCapitalR]\^n\ up\ to\ scalar\ \ *) \)\), " "}], "\[IndentingNewLine]", RowBox[{\(\(f[x_List] := 1/Sqrt[Sum[x[\([j]\)]^2, {j, 1, Length[x]}]]^ Length[x]\ x\)\(\[IndentingNewLine]\) \( (*\ Newton\ potential\ up\ on\ \[DoubleStruckCapitalR]\^n\ to\ scalar\ *) \ \)\), " "}], "\[IndentingNewLine]", RowBox[{\(g[x_List] := 1/\((2 - Length[x])\) 1/Sqrt[Sum[x[\([j]\)]^2, {j, 1, Length[x]}]]^\((Length[x] - 2)\)\), "\[IndentingNewLine]", \( (*\ vector\ in\ \[DoubleStruckCapitalR]\^n\ *) \)}], \ "\[IndentingNewLine]", RowBox[{\(co[x_, n_] := Table[x\_j, {j, n}]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \ \(Print["\"]\), "\[IndentingNewLine]", \(n = 3\), "\[IndentingNewLine]", \(xv = co[x, n];\), "\[IndentingNewLine]", \(f[ xv]\), "\[IndentingNewLine]", \(p[f, xv]\), "\[IndentingNewLine]", \(d[ f, xv]\), "\[IndentingNewLine]", \(g[xv]\), "\[IndentingNewLine]", \(gr[ g, xv]\), "\[IndentingNewLine]", RowBox[{\(% == f[xv]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \ \(Print["\"]\), "\[IndentingNewLine]", \(n = 5\), "\[IndentingNewLine]", \(xv = co[x, n];\), "\[IndentingNewLine]", \(f[ xv]\), "\[IndentingNewLine]", \(p[f, xv]\), "\[IndentingNewLine]", \(d[ f, xv]\), "\[IndentingNewLine]", \(g[xv]\), "\[IndentingNewLine]", \(gr[ g, xv]\), "\[IndentingNewLine]", RowBox[{\(% == f[xv]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \ \(Print["\"]\), "\[IndentingNewLine]", \(n = 2\), "\[IndentingNewLine]", \(xv = co[x, n];\), "\[IndentingNewLine]", \(f[ xv]\), "\[IndentingNewLine]", \(p[f, xv]\), "\[IndentingNewLine]", \(d[ f, xv]\), "\[IndentingNewLine]", \(g[{x1_, x2_}] := 1/2 Log[x1^2 + x2^2]\), "\[IndentingNewLine]", \(g[ xv]\), "\[IndentingNewLine]", \(gr[g, xv]\), "\[IndentingNewLine]", \(% == f[xv]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 7.9.3 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(<< Calculus`VectorAnalysis`\), "\[IndentingNewLine]", \(<< LinearAlgebra`Orthogonalization`\), "\[IndentingNewLine]", \(f[{x1_, x2_, x3_}] := {x1\ x3^2, x1^2\ x2 - x3^3, 2 x1\ x2 + x2^2 x3}\), "\[IndentingNewLine]", \(\(\(Div[f[{x1, x2, x3}], Cartesian[x1, x2, x3]] /. Thread[{x1, x2, x3} \[Rule] {x\_1, x\_2, x\_3}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := a {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]]}\), "\[IndentingNewLine]", \(MatrixForm[\[Phi][\[Alpha], \[Theta]]]\), "\[IndentingNewLine]", \(\[Omega]\_\[Phi][\[Alpha]_, \[Theta]_] := Simplify[Norm[ Cross[\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]], \ \[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]]]], {a > 0, Im[\[Alpha]] \[Equal] 0, 0 < \[Theta] < \[Pi]/2}]\), "\[IndentingNewLine]", \(\[Omega]\_\[Phi][\[Alpha], \[Theta]]\), "\[IndentingNewLine]", \(n[\[Alpha]_, \[Theta]_] := Simplify[Normalize[ Cross[\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]], \ \[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]]]], {a > 0, 0 < \[Theta] < \[Pi]/2}]\), "\[IndentingNewLine]", \(MatrixForm[n[\[Alpha], \[Theta]]]\), "\[IndentingNewLine]", \(FullSimplify[ f[\[Phi][\[Alpha], \[Theta]]] . n[\[Alpha], \[Theta]] \[Omega]\_\[Phi][\[Alpha], \[Theta]]]\), "\ \[IndentingNewLine]", \(\(\(Integrate[ f[\[Phi][\[Alpha], \[Theta]]] . n[\[Alpha], \[Theta]] \[Omega]\_\[Phi][\[Alpha], \[Theta]], {\ \[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], 0, \[Pi]/2}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(Table[ Integrate[ Evaluate[ Norm[{x1, x2, x3}]^2 Boole[Norm[{x1, x2, x3}] < j]], {x1, \(-j\), j}, {x2, \(-j\), j}, {x3, 0, j}], {j, 3}]\), "\[IndentingNewLine]", \(% \[Equal] 2\ \[Pi]\ /5 {1^5, 2^5, 3^5}\), "\[IndentingNewLine]", \(n[y1_, y2_] := {0, 0, 1}\), "\[IndentingNewLine]", \(Table[ Integrate[ f[{y1, y2, 0}] . n[y1, y2] Boole[Norm[{y1, y2}] < j], {y1, \(-j\), j}, {y2, \(-j\), j}], {j, 3}]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 8: THEORY", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Lemma 8.1.5 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\((j = {{0, \(-1\)}, {1, 0}})\) // MatrixForm\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_] := {Cos[\[Alpha]], Sin[\[Alpha]]}\), "\[IndentingNewLine]", \(\(\[Alpha] = \[Pi]/3;\)\), "\[IndentingNewLine]", \(\(c = ParametricPlot[\[Phi][\[Alpha]], {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] darkblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(l1 = Graphics[{darkblue, Line[{0 \[Phi][\[Alpha]], \[Phi][\[Alpha]]}], Line[{0 \[Phi][\[Alpha]], j . \[Phi][\[Alpha]]}]}];\)\), "\[IndentingNewLine]", \(\(p1 = Graphics[{RGBColor[1, 0, 0], PointSize[0.025], Point[\[Phi][\[Alpha]]]}];\)\), "\[IndentingNewLine]", \(\(p2 = Graphics[{darkblue, PointSize[0.025], Point[0 \[Phi][\[Alpha]]], Point[j . \[Phi][\[Alpha]]]}];\)\), "\[IndentingNewLine]", \(\(g1 = Show[c, l1, p1, p2, AspectRatio \[Rule] Automatic, Axes \[Rule] False, Background \[Rule] lightblue];\)\), "\[IndentingNewLine]", \(\(\[Alpha] = 2 \[Pi]/3;\)\), "\[IndentingNewLine]", \(\(l2 = Graphics[{darkblue, Line[{0 \[Phi][\[Alpha]], \[Phi][\[Alpha]]}], Line[{0 \[Phi][\[Alpha]], j . \[Phi][\[Alpha]]}]}];\)\), "\[IndentingNewLine]", \(\(p3 = Graphics[{RGBColor[1, 0, 0], PointSize[0.025], Point[\[Phi][\[Alpha]]]}];\)\), "\[IndentingNewLine]", \(\(p4 = Graphics[{darkblue, PointSize[0.025], Point[0 \[Phi][\[Alpha]]], Point[j . \[Phi][\[Alpha]]]}];\)\), "\[IndentingNewLine]", \(\(g2 = Show[c, l2, p3, p4, AspectRatio \[Rule] Automatic, Axes \[Rule] False, Background \[Rule] lightblue];\)\), "\[IndentingNewLine]", \(\(\(Show[GraphicsArray[{g1, g2}], AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction -> $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Do[task = setdel[Format[\(\(Derivative[KroneckerDelta[j, 1], KroneckerDelta[j, 2]]\)[Subscript[f_, i_]]\)[k__]], TraditionalForm[\(D\_j\) f\_i[k]]]; task /. setdel \[Rule] SetDelayed, {j, 2}]\), "\[IndentingNewLine]", \(jacobimatrix[f_, x_List] := \ With[{y = \(Unique[y] &\) /@ x}, Table[D[\(f[y]\)[\([i]\)], y[\([j]\)]], {i, Length[f[y]]}, {j, Length[x]}] /. \ Thread[y \[Rule] x]]\), "\[IndentingNewLine]", \(f[{x1_, x2_}] := {f\_1[x1, x2], f\_2[x1, x2]}\), "\[IndentingNewLine]", \(co[x_, n_] := Table[x\_j, {j, n}]\), "\[IndentingNewLine]", \(\(xv = co[x, 2];\)\), "\[IndentingNewLine]", \(\(\(t[f_]\)[x_List] = Transpose[j] . f[x];\)\), "\[IndentingNewLine]", \(\(t[f]\)[xv]\), "\[IndentingNewLine]", \(\(\(Tr[jacobimatrix[t[f], xv]] j\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(r[{x1_, x2_}] := j . {x1, x2}\), "\[IndentingNewLine]", \(r[xv]\), "\[IndentingNewLine]", \(\(t[r]\)[xv]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Example 8.1.7 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(<< Graphics`PlotField`\), "\[IndentingNewLine]", \(Off[Power::infy]\), "\[IndentingNewLine]", \(Off[\[Infinity]::indet]\), "\[IndentingNewLine]", \(f[x1_, x2_] := {\(-x2\), x1}\), "\[IndentingNewLine]", \(\(PlotVectorField[f[x1, x2], {x1, \(-1\), 1}, {x2, \(-1\), 1}, ScaleFactor \[Rule] 0.5, PlotPoints \[Rule] 50, ImageSize \[Rule] 400, ColorFunction \[Rule] Hue, Background \[Rule] lightblue];\)\), "\[IndentingNewLine]", \(f[x1_, x2_] := {\(-x2\), x1}\), "\[IndentingNewLine]", \(f[x1_, x2_] := \(1\/\(x1\^2 + x2\^2\)\) {\(-x2\), x1}\), "\[IndentingNewLine]", \(\(PlotVectorField[f[x1, x2], {x1, \(-1\), 1}, {x2, \(-1\), 1}, ScaleFactor \[Rule] 0.5, PlotPoints \[Rule] 50, ImageSize \[Rule] 400, ColorFunction \[Rule] Hue, Background \[Rule] lightblue];\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Lemma 8.1.8 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(co[x_, n_] := Table[x\_j, {j, n}]\), "\[IndentingNewLine]", \(\(av = co[a, 3];\)\), "\[IndentingNewLine]", \(\(hv = co[h, 3];\)\), "\[IndentingNewLine]", \(\(kv = co[k, 3];\)\), "\[IndentingNewLine]", \(\((anti = {{0, \(-a\_3\), a\_2}, {a\_3, 0, \(-a\_1\)}, {\(-a\_2\), a\_1, 0}})\) // MatrixForm\), "\[IndentingNewLine]", \(anti . hv \[Equal] Cross[av, hv]\), "\[IndentingNewLine]", \(\((anti . hv)\) . kv \[Equal] av . Cross[hv, kv]\), "\[IndentingNewLine]", \(Simplify[\((anti . hv)\) . kv \[Equal] av . Cross[hv, kv]]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Corollary 8.1.10", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(Do[task = setdel[Format[\(\(Derivative[KroneckerDelta[j, 1], KroneckerDelta[j, 2], KroneckerDelta[j, 3]]\)[ Subscript[f_, i_]]\)[k__]], TraditionalForm[\(D\_j\) f\_i[k]]]; task /. setdel \[Rule] SetDelayed, {j, 3}]\), "\[IndentingNewLine]", \(jacobimatrix[f_, x_List] := \ With[{y = \(Unique[y] &\) /@ x}, Table[D[\(f[y]\)[\([i]\)], y[\([j]\)]], {i, Length[f[y]]}, {j, Length[x]}] /. \ Thread[y \[Rule] x]]\), "\[IndentingNewLine]", \(f[{x1_, x2_, x3_}] := {f\_1[x1, x2, x3], f\_2[x1, x2, x3], f\_3[x1, x2, x3]}\), "\[IndentingNewLine]", \(co[x_, n_] := Table[x\_j, {j, n}]\), "\[IndentingNewLine]", \(\(xv = co[x, 3];\)\), "\[IndentingNewLine]", \(\(hv = co[h, 3];\)\), "\[IndentingNewLine]", \(\(kv = co[k, 3];\)\), "\[IndentingNewLine]", \(antif = \(\(jacobimatrix[f, xv]\)\(-\)\(Transpose[ jacobimatrix[f, xv]]\)\(\ \)\)\), "\[IndentingNewLine]", \(i[j_] := Mod[j, 3, 1]\), "\[IndentingNewLine]", \(\(\(curlf = Table[antif[\([i[j + 2], i[j + 1]]\)], {j, 1, 3}]\)\(\ \)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(<< Calculus`VectorAnalysis`\), "\[IndentingNewLine]", \(Curl[f[{x1, x2, x3}], Cartesian[x1, x2, x3]] /. Thread[{x1, x2, x3} \[Rule] {x\_1, x\_2, x\_3}]\), "\[IndentingNewLine]", \(curlf \[Equal] %\), "\[IndentingNewLine]", \(Simplify[ antif . hv \[Equal] Cross[curlf, hv]]\), "\[IndentingNewLine]", \(Simplify[\((antif . hv)\) . kv \[Equal] curlf . Cross[hv, kv]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 8.2.4", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\((j = {{0, \(-1\)}, {1, 0}})\) // MatrixForm\), "\[IndentingNewLine]", \(f[{x1_, x2_}] := 1/\((x1^2 + x2^2)\) j . {x1, x2}\), "\[IndentingNewLine]", \(\(xv = {x\_1, x\_2};\)\), "\[IndentingNewLine]", \(f[xv]\), "\[IndentingNewLine]", \(\[Gamma][t_] := {Cos[t], Sin[t]}\), "\[IndentingNewLine]", \(\(\[Gamma]'\)[t]\), "\[IndentingNewLine]", \(f[\[Gamma][t]]\), "\[IndentingNewLine]", \(Simplify[f[\[Gamma][t]] . \(\[Gamma]'\)[t]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Section 8.3 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\), "\[IndentingNewLine]", \(\(j = {{0, \(-1\)}, {1, 0}};\)\), "\[IndentingNewLine]", \(y[t_] := {Cos[t], Sin[t]}\), "\[IndentingNewLine]", \(y[t] // MatrixForm\), "\[IndentingNewLine]", \(j . y[t] == \(y'\)[t]\), "\[IndentingNewLine]", \(\(\(Simplify[Det[{y[t], \(y'\)[t]}]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(y[t_] := 3/\((1 + t^3)\) {t, t^2}\), "\[IndentingNewLine]", \(tt[t_] := 1/Norm[\(y'\)[t]] \(y'\)[t]\), "\[IndentingNewLine]", \(\(j = {{0, \(-1\)}, {1, 0}};\)\), "\[IndentingNewLine]", \(nn[t_] := 1/Norm[\(y'\)[t]] Transpose[j] . \(y'\)[t]\), "\[IndentingNewLine]", \(Simplify[j . nn[t] \[Equal] tt[t]]\), "\[IndentingNewLine]", \(\(\(FullSimplify[Det[{nn[t], tt[t]}], t > 0]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[Evaluate[y[t]], {t, 0, 100}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[Evaluate[y[t]], {t, 0, 100}, PlotStyle \[Rule] lightgreen, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = p2 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(l1 = Graphics[{RGBColor[1, 0, 0], Line[{{0, 0}, {0, 0.03}}]}];\)\), "\[IndentingNewLine]", \(\(t = 3/2;\)\), "\[IndentingNewLine]", \(\(l2 = Graphics[{darkblue, Line[{y[t], y[t] + tt[t]}]}];\)\), "\n", \(\(l3 = Graphics[{darkblue, Line[{y[t], y[t] + nn[t]}]}];\)\), "\[IndentingNewLine]", \(\(a1 = Graphics[{darkblue, PointSize[0.013], Point[y[t]], Point[y[t] + tt[t]], Point[y[t] + nn[t]]}];\)\), "\[IndentingNewLine]", \(\(Show[p3, p1, l1, l2, l3, a1, PlotRange \[Rule] {0.5, 2.7}, Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 8.3.2 (Annular domain)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\), "\[IndentingNewLine]", \(\(\(j = {{0, \(-1\)}, {1, 0}};\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(y1[t_] := 4 {Cos[t], Sin[t]}\), "\[IndentingNewLine]", \(y2[t_] := 2 {Cos[t], \(-Sin[t]\)}\), "\[IndentingNewLine]", \(tt1[t_] := 1/Norm[\(y1'\)[t]] \(y1'\)[t]\), "\[IndentingNewLine]", \(nn1[t_] := 1/Norm[\(y1'\)[t]] Transpose[j] . \(y1'\)[t]\), "\[IndentingNewLine]", \(Simplify[j . nn1[t] \[Equal] tt1[t]]\), "\[IndentingNewLine]", \(FullSimplify[Det[{nn1[t], tt1[t]}], t > 0]\), "\[IndentingNewLine]", \(tt2[t_] := 1/Norm[\(y2'\)[t]] \(y2'\)[t]\), "\[IndentingNewLine]", \(nn2[t_] := 1/Norm[\(y2'\)[t]] Transpose[j] . \(y2'\)[t]\), "\[IndentingNewLine]", \(Simplify[j . nn2[t] \[Equal] tt2[t]]\), "\[IndentingNewLine]", \(\(\(FullSimplify[Det[{nn2[t], tt2[t]}], t > 0]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[Evaluate[y1[t]], {t, \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[Evaluate[y1[t]], {t, \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] lightgreen, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = p2 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot[Evaluate[y2[t]], {t, \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p5 = ParametricPlot[Evaluate[y2[t]], {t, \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] lightblue, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p6 = p5 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(t = 2 \[Pi]/3;\)\), "\[IndentingNewLine]", \(\(g1 = Graphics[{darkblue, PointSize[0.011], Point[y1[t]], Point[y1[t] + tt1[t]], Point[y1[t] + nn1[t]]}];\)\), "\[IndentingNewLine]", \(\(g2 = Graphics[{darkblue, Line[{y1[t], y1[t] + tt1[t]}], Line[{y1[t], y1[t] + nn1[t]}]}];\)\), "\[IndentingNewLine]", \(\(t = \[Pi]/3;\)\), "\[IndentingNewLine]", \(\(g3 = Graphics[{darkblue, Line[{y2[t], y2[t] + tt2[t]}], Line[{y2[t], y2[t] + nn2[t]}]}];\)\), "\n", \(\(g4 = Graphics[{darkblue, PointSize[0.011], Point[y2[t]], Point[y2[t] + tt2[t]], Point[y2[t] + nn2[t]]}];\)\), "\[IndentingNewLine]", \(\(\(Show[p3, p6, p1, p4, g1, g2, g3, g4, Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(Clear[p1, p2, p3, p4, p5, p6, l, g1, g2, g3, g4, t];\)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[Evaluate[y1[t]], {t, 0, \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[Evaluate[y1[t]], {t, 0, \[Pi]}, PlotStyle \[Rule] lightgreen, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = p2 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot[Evaluate[y2[t]], {t, \(-\[Pi]\), 0}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p5 = ParametricPlot[Evaluate[y2[t]], {t, \(-\[Pi]\), 0}, PlotStyle \[Rule] lightblue, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p6 = p5 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(l = Graphics[{RGBColor[1, 0, 0], Line[{y1[\[Pi]], y2[\(-\[Pi]\)]}], Line[{y1[0], y2[0]}]}];\)\), "\[IndentingNewLine]", \(\(t = 2 \[Pi]/3;\)\), "\[IndentingNewLine]", \(\(g1 = Graphics[{darkblue, PointSize[0.011], Point[y1[t]], Point[y1[t] + tt1[t]], Point[y1[t] + nn1[t]]}];\)\), "\[IndentingNewLine]", \(\(g2 = Graphics[{darkblue, Line[{y1[t], y1[t] + tt1[t]}], Line[{y1[t], y1[t] + nn1[t]}]}];\)\), "\[IndentingNewLine]", \(\(t = \(-2\) \[Pi]/3;\)\), "\[IndentingNewLine]", \(\(g3 = Graphics[{darkblue, Line[{y2[t], y2[t] + tt2[t]}], Line[{y2[t], y2[t] + nn2[t]}]}];\)\), "\n", \(\(g4 = Graphics[{darkblue, PointSize[0.011], Point[y2[t]], Point[y2[t] + tt2[t]], Point[y2[t] + nn2[t]]}];\)\), "\[IndentingNewLine]", \(\(g5 = Graphics[{darkblue, Line[{{2.5, 0}, {2.5, \(-1\)}}], Line[{{2.5, 0}, {3.5, 0}}], Line[{{\(-3.5\), 0}, {\(-3.5\), \(-1\)}}], Line[{{\(-3.5\), 0}, {\(-2.5\), 0}}]}];\)\), "\[IndentingNewLine]", \(\(g6 = Graphics[{darkblue, PointSize[0.011], Point[{2.5, 0}], Point[{2.5, \(-1\)}], Point[{3.5, 0}], Point[{\(-3.5\), 0}], Point[{\(-3.5\), \(-1\)}], Point[{\(-2.5\), 0}]}];\)\), "\[IndentingNewLine]", \(\(\(Show[p3, p6, l, p1, p4, g1, g2, g3, g4, g5, g6, Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(Clear[p1, p2, p3, p4, p5, p6, l, g1, g2, g3, g4, t];\)\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[Evaluate[y1[t]], {t, \(-\[Pi]\), 0}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[Evaluate[y1[t]], {t, \(-\[Pi]\), 0}, PlotStyle \[Rule] lightgreen, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = p2 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(p4 = ParametricPlot[Evaluate[y2[t]], {t, 0, \[Pi]}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p5 = ParametricPlot[Evaluate[y2[t]], {t, 0, \[Pi]}, PlotStyle \[Rule] lightblue, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p6 = p5 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(l = Graphics[{RGBColor[1, 0, 0], Line[{y1[\[Pi]], y2[\(-\[Pi]\)]}], Line[{y1[0], y2[0]}]}];\)\), "\[IndentingNewLine]", \(\(t = \(-\[Pi]\)/3;\)\), "\[IndentingNewLine]", \(\(g1 = Graphics[{darkblue, PointSize[0.011], Point[y1[t]], Point[y1[t] + tt1[t]], Point[y1[t] + nn1[t]]}];\)\), "\[IndentingNewLine]", \(\(g2 = Graphics[{darkblue, Line[{y1[t], y1[t] + tt1[t]}], Line[{y1[t], y1[t] + nn1[t]}]}];\)\), "\[IndentingNewLine]", \(\(t = \[Pi]/3;\)\), "\[IndentingNewLine]", \(\(g3 = Graphics[{darkblue, Line[{y2[t], y2[t] + tt2[t]}], Line[{y2[t], y2[t] + nn2[t]}]}];\)\), "\n", \(\(g4 = Graphics[{darkblue, PointSize[0.011], Point[y2[t]], Point[y2[t] + tt2[t]], Point[y2[t] + nn2[t]]}];\)\), "\[IndentingNewLine]", \(\(g5 = Graphics[{darkblue, Line[{{3.5, 0}, {3.5, 1}}], Line[{{2.5, 0}, {3.5, 0}}], Line[{{\(-2.5\), 0}, {\(-2.5\), 1}}], Line[{{\(-3.5\), 0}, {\(-2.5\), 0}}]}];\)\), "\[IndentingNewLine]", \(\(g6 = Graphics[{darkblue, PointSize[0.011], Point[{2.5, 0}], Point[{3.5, 1}], Point[{3.5, 0}], Point[{\(-3.5\), 0}], Point[{\(-2.5\), 1}], Point[{\(-2.5\), 0}]}];\)\), "\[IndentingNewLine]", \(\(Show[p3, p6, l, p1, p4, g1, g2, g3, g4, g5, g6, Axes \[Rule] False, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 8.3.6 (Descartes' folium) Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(y[t_] := 3 a/\((1 + t^3)\) {t, t^2}\), "\[IndentingNewLine]", \(y1[t_] := \(y[t]\)[\([1]\)]\), "\n", \(y2[t_] := \(y[t]\)[\([2]\)]\), "\[IndentingNewLine]", \(i[t_] = Simplify[y1[t] \(y2'\)[t] - y2[t] \(y1'\)[t]]\), "\n", \(\(\(Integrate[1/2 i[t], {t, 0, \[Infinity]}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(a = 1;\)\), "\n", \(\(p1 = ParametricPlot[Evaluate[y[t]], {t, 0, 100}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-0.05\), 1.65}, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[Evaluate[y[t]], {t, 0, 100}, PlotStyle \[Rule] lightgreen, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-0.05\), 1.65}, PlotPoints \[Rule] 200, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = p2 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(l = Graphics[{RGBColor[1, 0, 0], Line[{{0, 0}, {0, 0.03}}]}];\)\), "\[IndentingNewLine]", \(\(Show[p3, p1, l, Background \[Rule] lightblue, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Lemma 8.4.2", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(\(\(Clear["\<`*\>"]\)\(\[IndentingNewLine]\) \) (*\ formatting\ partial\ derivatives\ *) \ \), "\[IndentingNewLine]", \(\(Format[\(\(Derivative[l__]\)[f_]\)[ x___]] := \(D\_\(First@First[Position[List[l], j_ /; j > 0]]\)\) f[x];\)\[IndentingNewLine] (*\ formatting\ Jacobi\ matrices\ *) \), "\[IndentingNewLine]", \(\(\(jacobimatrix[f_, x_List] := \ With[{y = \(Unique[y] &\) /@ x}, Table[\[PartialD]\_\(y[\([j]\)]\)\ \(f[y]\)[\([i]\)], {i, Length[f[y]]}, {j, Length[x]}] /. \ Thread[y \[Rule] x]]\)\(\[IndentingNewLine]\) \) (*\ column\ vector\ of\ Jacobi\ matrix\ *) \ \), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(\(\(d[j_]\)[f_, x_List] := \ With[{y = \(Unique[y] &\) /@ x}, Table[\[PartialD]\_\(y[\([j]\)]\)\ \(f[y]\)[\([i]\)], {i, Length[f[y]]}] /. \ Thread[y \[Rule] x]]\)\(\[IndentingNewLine]\) \( (*\ antisymmetrized\ Jacobi\ matrix\ *) \)\)\), "\[IndentingNewLine]", \(\(\(anti[f_, x_List] := jacobimatrix[f, x] - Transpose[jacobimatrix[f, x]]\)\(\ \)\(\[IndentingNewLine]\) \) (*\ curl\ of\ vector\ field\ in\ \[DoubleStruckCapitalR]\^3\ *) \ \), "\ \[IndentingNewLine]", \(i[j_] := Mod[j, 3, 1]\), "\[IndentingNewLine]", \(\(\(curl3[f_, x_List] := Table[\(anti[f, x]\)[\([i[j + 2], i[j + 1]]\)], {j, Length[x]}]\)\(\ \)\(\[IndentingNewLine]\) \) (*\ curl\ of\ vector\ field\ in\ \[DoubleStruckCapitalR]\^2\ *) \ \), "\ \[IndentingNewLine]", \(curl2[g_, y_List] := \(anti[g, y]\)[\([2, 1]\)]\), "\[IndentingNewLine]", \(\(\(co[x_, n_] := Table[x\_j, {j, n}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(xv = co[x, 3];\)\), "\[IndentingNewLine]", \(\(yv = co[y, 2];\)\), "\[IndentingNewLine]", \(f[{x1_, x2_, x3_}] := {f\_1[x1, x2, x3], f\_2[x1, x2, x3], f\_3[x1, x2, x3]}\), "\[IndentingNewLine]", \(\(\(\[Phi][{y1_, y2_}] := {\[Phi]\_1[y1, y2], \[Phi]\_2[y1, y2], \[Phi]\_3[y1, y2]}\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(f[xv]\), "\[IndentingNewLine]", \(curl3[f, xv]\), "\[IndentingNewLine]", \(\[Phi][yv]\), "\[IndentingNewLine]", \(f[\[Phi][yv]]\), "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(g[y_] := Transpose[jacobimatrix[\[Phi], y]] . f[\[Phi][y]]\), "\[IndentingNewLine]", \(g[yv] // MatrixForm\), "\[IndentingNewLine]", \(c1 = curl2[g, yv]\), "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(curl3[f, \[Phi][yv]]\), "\[IndentingNewLine]", \(cr = Cross[\(d[1]\)[\[Phi], yv], \(d[2]\)[\[Phi], yv]]\), "\[IndentingNewLine]", \(c2 = curl3[f, \[Phi][yv]] . cr\), "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(Simplify[c1 == c2]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Remark on compatible orientations ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(rmin = 0;\)\), "\n", \(\(rmax = 0.5;\)\), "\n", \(\(\[Alpha]min = \(-\[Pi]\);\)\), "\n", \(\(\[Alpha]max = \[Pi];\)\), "\[IndentingNewLine]", \(\(nr = 59;\)\), "\n", \(\(n\[Alpha] = 99;\)\), "\n", \(f[x1_, x2_] := x1\^2 - x1\ x2\^3\), "\[IndentingNewLine]", \(x1[r_, \[Alpha]_] := r\ Cos[\[Alpha]]\), "\n", \(x2[r_, \[Alpha]_] := r\ Sin[\[Alpha]]\), "\n", \(x3[r_, \[Alpha]_] := f[x1[r, \[Alpha]], x2[r, \[Alpha]]]\), "\[IndentingNewLine]", \(x[r_, \[Alpha]_] := {x1[r, \[Alpha]], x2[r, \[Alpha]], x3[r, \[Alpha]]}\), "\n", \(\(g = ParametricPlot3D[ Append[x[r, \[Alpha]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {r, rmin, rmax}, {\[Alpha], \[Alpha]min, \[Alpha]max}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {nr, n\[Alpha]}, Boxed \[Rule] False, Axes \[Rule] False, Ticks \[Rule] None, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(p1 = x[0.5, \[Pi]];\)\), "\[IndentingNewLine]", \(\(t1 = {\[PartialD]\_\[Alpha]\ x1[r, \[Alpha]], \[PartialD]\_\[Alpha]\ x2[r, \[Alpha]], \[PartialD]\_\[Alpha]\ x3[r, \[Alpha]]} /. {r \[Rule] 0.5, \[Alpha] \[Rule] \[Pi]};\)\), "\[IndentingNewLine]", \(\(c1 = Cross[{\[PartialD]\_r\ x1[r, \[Alpha]], \[PartialD]\_r\ x2[r, \[Alpha]], \[PartialD]\_r\ x3[r, \[Alpha]]}, {\[PartialD]\_\[Alpha]\ x1[r, \[Alpha]], \[PartialD]\_\[Alpha]\ x2[r, \[Alpha]], \[PartialD]\_\[Alpha]\ x3[r, \[Alpha]]}] /. {r \[Rule] 0.5, \[Alpha] \[Rule] \[Pi]};\)\), "\[IndentingNewLine]", \(\(n1 = 0.5/Norm[c1]\ c1;\)\), "\[IndentingNewLine]", \(\(l1 = Graphics3D[{darkblue, Thickness[0.004], Line[{p1, p1 + Cross[t1, n1]}]}];\)\), "\[IndentingNewLine]", \(\(l2 = Graphics3D[{RGBColor[1, 0, 0], Thickness[0.004], Line[{p1, p1 + 0.5 t1}]}];\)\), "\[IndentingNewLine]", \(\(l3 = Graphics3D[{RGBColor[0, 0.5, 0], Thickness[0.004], Line[{p1, p1 + 0.5 n1}]}];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(p2 = x[0.5, 0];\)\), "\[IndentingNewLine]", \(\(t2 = {\[PartialD]\_\[Alpha]\ x1[r, \[Alpha]], \[PartialD]\_\[Alpha]\ x2[r, \[Alpha]], \[PartialD]\_\[Alpha]\ x3[r, \[Alpha]]} /. {r \[Rule] 0.5, \[Alpha] \[Rule] 0};\)\), "\[IndentingNewLine]", \(\(c2 = Cross[{\[PartialD]\_r\ x1[r, \[Alpha]], \[PartialD]\_r\ x2[r, \[Alpha]], \[PartialD]\_r\ x3[r, \[Alpha]]}, {\[PartialD]\_\[Alpha]\ x1[r, \[Alpha]], \[PartialD]\_\[Alpha]\ x2[r, \[Alpha]], \[PartialD]\_\[Alpha]\ x3[r, \[Alpha]]}] /. {r \[Rule] 0.5, \[Alpha] \[Rule] 0};\)\), "\[IndentingNewLine]", \(\(n2 = 0.5/Norm[c2]\ c2;\)\), "\[IndentingNewLine]", \(\(l4 = Graphics3D[{darkblue, Thickness[0.004], Line[{p2, p2 + Cross[t2, n2]}]}];\)\), "\[IndentingNewLine]", \(\(l5 = Graphics3D[{RGBColor[1, 0, 0], Thickness[0.004], Line[{p2, p2 + 0.5 t2}]}];\)\), "\[IndentingNewLine]", \(\(l6 = Graphics3D[{RGBColor[0, 0.5, 0], Thickness[0.004], Line[{p2, p2 + 0.5 n2}]}];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(Print["\"]\), \ "\[IndentingNewLine]", \(\(Show[g, l1, l2, l3, l4, l5, l6, PlotRange \[Rule] All, Background \[Rule] lightblue, ViewPoint \[Rule] 3 {4, 2, 0.7}, ImageSize \[Rule] 1000, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example 8.5.1", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(\(Clear["\<`*\>"];\)\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(x1[a_, t_] := Cos[a] Cos[t];\)\), "\n", \(\(x2[a_, t_] := Sin[a] Cos[t];\)\), "\n", \(\(x3[a_, t_] := Sin[t];\)\), "\n", \(\(\(x[a_, t_] := {x1[a, t], x2[a, t], 1 + x3[a, t]};\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(raster = 0.04;\)\), "\n", \(\(point = 0.001;\)\), "\n", \(\(list1 = Table[Point[x[a, t]], {a, \(-\[Pi]\), \[Pi], raster}, {t, \(-\(\[Pi]\/2\)\) + 0.01, \[Pi]\/2 - 0.01, raster}];\)\), "\n", \(\(sphere = Graphics3D[{PointSize[point], darkblue, list1}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(n = Graphics3D[{RGBColor[1, 0, 0], PointSize[0.01], Point[x[0, \[Pi]\/2]]}];\)\[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(\[Psi] = \(-\(\[Pi]\/10\)\);\)\), "\[IndentingNewLine]", \(\[Gamma][\[Alpha]_] := x[\[Alpha], \[Psi]]\), "\n", \(\(circ = ParametricPlot3D[ Evaluate[\[Gamma][\[Alpha]]], {\[Alpha], \(-\[Pi]\), \[Pi]}, Axes \[Rule] False, PlotRange \[Rule] All, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine]\), "\n", \(\[Mu][\[Theta]_] := x[\[Pi], \[Theta]]\), "\n", \(\(mer = ParametricPlot3D[ Evaluate[\[Mu][\[Theta]]], {\[Theta], \[Psi], \[Pi]\/2}, Axes \[Rule] False, AspectRatio \[Rule] 1, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\n\), "\[IndentingNewLine]", \(\(Show[sphere, circ, mer, n, ViewPoint \[Rule] 3 {3, 1.5, 1}, AspectRatio \[Rule] Automatic, ImageSize \[Rule] 900, Background \[Rule] lightblue, DefaultColor \[Rule] RGBColor[1, 0, 0], DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example 8.5.3 Illustration\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(moebius[t_, a_] := {Cos[a] \((1 + t\ Cos[a/2]\ )\), Sin[a] \((1 + t\ Cos[a/2]\ )\), t\ Sin[a/2]}\), "\n", \(\(ParametricPlot3D[ Append[moebius[t, a], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {a, 0, 2 \[Pi]}, {t, \(- .5\), .5}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {80, 10}, Axes \[Rule] None, Boxed \[Rule] False, ViewPoint \[Rule] {0, \(-4\), 3}, AspectRatio \[Rule] 0.6, Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 6: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 6.2 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[1\/\((x1 + x2)\)\^2, {x1, 0, 1}, {x2, 1, 2}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.3 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\), "\[IndentingNewLine]", \(Solve[x2^2 == 2 x2, x2] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]", \(<< Graphics`ImplicitPlot`\), "\[IndentingNewLine]", \(\(p1 = ImplicitPlot[x1 == x2, {x1, 0, 2.5}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ImplicitPlot[x2^2 \[Equal] 2 x1, {x1, 0, 2.5}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = ImplicitPlot[x2^2 \[Equal] 2 x1, {x1, 0, 2}, PlotStyle \[Rule] lightgreen, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p4 = p3 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(Show[p4, p1, p2, PlotRange \[Rule] {0, 2.5}, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction -> $DisplayFunction];\)\), "\[IndentingNewLine]", \(Integrate[ Boole[x2^2/2 \[LessEqual] x1 \[LessEqual] x2], {x2, 0, 2}, {x1, 0, 2}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.4 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Integrate[ x1^2 + x2^2, {x1, 0, 1}, {x2, 0, 1}]\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[{r\ Cos[\[Alpha]], r\ Sin[\[Alpha]], r^2}, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {r, 0, 2}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {121, 121}, ViewPoint \[Rule] {\(-1\), \(-2\), 0.6}, PlotRange \[Rule] {{0, 1}, {0, 1}, {0, 2}}, Ticks \[Rule] False, BoxStyle \[Rule] RGBColor[0, 0.5, 0], BoxRatios \[Rule] {1, 1, 1}, Boxed \[Rule] True, Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.6", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(a = 3;\)\), "\n", \(\(p1 = Plot3D[x^2 + y^2, {x, \(-4\), 4}, {y, \(-4\), 4}, DefaultColor \[Rule] darkblue, LightSources \[Rule] {{{1, 0, \(-1\)}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] 60, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = ParametricPlot3D[ Append[{a\ Cos[t], a\ Sin[t], s}, SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {t, 0, 2\ \[Pi]}, {s, 0, 30}, LightSources \[Rule] {{{1, 0, \(-1\)}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] {80, 30}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(\(Show[p1, p2, ViewPoint \[Rule] {\(-30\), \(-10\), \(-10\)}, Boxed \[Rule] False, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 700, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ Boole[0 \[LessSlantEqual] x3 \[LessSlantEqual] x1^2 + x2^2 \[LessEqual] a^2], {x1, \(-a\), a}, {x2, \(-a\), a}, {x3, 0, a^2}]\), "\[IndentingNewLine]", \(\(a = 4;\)\), "\[IndentingNewLine]", \(\(\(Integrate[ Boole[0 \[LessSlantEqual] x3 \[LessSlantEqual] x1^2 + x2^2 \[LessEqual] a^2], {x1, \(-a\), a}, {x2, \(-a\), a}, {x3, 0, a^2}]\)\(\[IndentingNewLine]\) \)\), "\n", \(Print["\"]\), "\[IndentingNewLine]", \(Clear[a]\), "\n", \(\(f[x1_, x2_] = Integrate[1, {x3, 0, x1^2 + x2^2}];\)\), "\n", \(f[x\_1, x\_2]\), "\n", \(\(f[x1_] = Integrate[f[x1, x2], {x2, \(-Sqrt[a^2 - x1^2]\), Sqrt[a^2 - x1^2]}, Assumptions \[Rule] {a > x1}];\)\), "\n", \(f[x\_1]\), "\n", \(Integrate[f[x1], {x1, \(-a\), a}, Assumptions \[Rule] {a > 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.7", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[ x1^2 x2^2 Boole[x1^2 + x2^2 \[LessEqual] 1], {x1, \(-1\), 1}, {x2, \(-1\), 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.8", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[ Boole[Norm[{x1, x2, x3}] \[LessSlantEqual] 1] x1\ x2\ x3\ Norm[{x1, x2, x3}], {x1, \(-1\), 1}, {x2, \(-1\), 1}, {x3, \(-1\), 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.9", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " detects a mistake in the book." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[N::meprec]\), "\[IndentingNewLine]", \(f[x1_, x2_] := 1/Sqrt[x1^2 + x2^2]\), "\[IndentingNewLine]", \(\(Plot3D[1/Sqrt[x1^2 + x2^2], {x1, \(-1\), 1}, {x2, \(-1\), 1}, DefaultColor \[Rule] RGBColor[0.7, 0.7, 1], LightSources \[Rule] {{{1, 0, 0}, RGBColor[0, 0, 1]}, {{\(-1\), 0, 0}, RGBColor[0, 0, 1]}, {{0, 1, 0}, RGBColor[0, 1, 1]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotRange \[Rule] {0, 100}, PlotPoints \[Rule] 120, ViewPoint \[Rule] {2, 0, 0.5}, Boxed \[Rule] False, Axes \[Rule] None, ImageSize \[Rule] 800];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ f[x1, x2], {x1, \(-1\), 1}, {x2, \(-1\), 1}]\), "\[IndentingNewLine]", \(a = 8 Log[1 + Sqrt[2]]\), "\[IndentingNewLine]", \(FullSimplify[%% \[Equal] a]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(f1[x1_, x2_] = Integrate[f[xx1, x2], xx1] /. xx1 \[Rule] x1;\)\), "\[IndentingNewLine]", \(f1[x\_1, x\_2]\), "\[IndentingNewLine]", \(\(g[x2_] := f1[1, x2] - f1[\(-1\), x2];\)\), "\[IndentingNewLine]", \(g[x\_2]\), "\[IndentingNewLine]", \(\(g1[x2_] := Integrate[g[xx2], xx2] /. xx2 \[Rule] x2;\)\), "\[IndentingNewLine]", \(g1[x\_2]\), "\[IndentingNewLine]", \(g1[1] - g1[\(-1\)]\), "\[IndentingNewLine]", \(FullSimplify[% \[Equal] a]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.10", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[ Boole[1 \[LessSlantEqual] x1 \[LessSlantEqual] \[ExponentialE]^x3 && x2 \[GreaterSlantEqual] x3 && x2^2 + x3^2 \[LessSlantEqual] 4]/ x1, {x1, 1, \[ExponentialE]^2}, {x2, \(-2\), 2}, {x3, \(-2\), 2}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.14 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Integrate[ Boole[x1^2/a^2 + x2^2/b^2 \[LessEqual] 1], {x1, \(-a\), a}, {x2, \(-b\), b}, Assumptions \[Rule] {a > 0, b > 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.15 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is not very skillful in handling these integrals. See also the file \ Ex6.15.pdf. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[\[ExponentialE]^\((\(-x^2\))\), {x, 0, a}]^2 == Integrate[ 2 \[ExponentialE]^\((\(-r^2\))\) r, {\[Alpha], 0, \[Pi]/4}, {r, 0, a/Cos[\[Alpha]]}], a > 0]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[ 2 \[ExponentialE]^\((\(-r^2\))\) r, {\[Alpha], 0, \[Pi]/4}, {r, 0, a/Cos[\[Alpha]]}] \[Equal] \[Pi]/4 - Integrate[\[ExponentialE]^\((\(-a^2\)/ Cos[\[Alpha]]^2)\), {\[Alpha], 0, \[Pi]/4}], a > 0]\), "\[IndentingNewLine]", \(FullSimplify[ Distribute[ Integrate[ 2 \[ExponentialE]^\((\(-r^2\))\) r, {\[Alpha], 0, \[Pi]/4}, {r, 0, a/Cos[\[Alpha]]}]] \[Equal] \[Pi]/4 - Integrate[\[ExponentialE]^\((\(-a^2\)/ Cos[\[Alpha]]^2)\), {\[Alpha], 0, \[Pi]/4}], a > 0]\), "\[IndentingNewLine]", \(f[a_] := Integrate[\[ExponentialE]^\((\(-x^2\))\), {x, 0, aa}]^2 + Integrate[\[ExponentialE]^\((\(-aa^2\) \((1 + t^2)\))\)/\((1 + t^2)\), {t, 0, 1}] /. aa \[Rule] a\), "\[IndentingNewLine]", \(FullSimplify[f[a], a > 0]\), "\[IndentingNewLine]", \(\[PartialD]\_a\ f[a]\), "\[IndentingNewLine]", \(f[0]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.16 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[N::meprec]\), "\[IndentingNewLine]", \(Simplify[ Integrate[ Boole[0 \[LessEqual] x1 \[LessEqual] 1 && 0 \[LessEqual] x2 \[LessEqual] 1 - x1] \[ExponentialE]^\((\((x1 - x2)\)/\((x1 + x2)\))\), {x1, 0, 1}, {x2, 0, 1}] \[Equal] Sinh[1]/2]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.17 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["In executing this exercise", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", FontSlant->"Italic"], " has been aborted after two hours of runtime." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[ \( (*\ Clear["\<`*\>"]\ \[IndentingNewLine]FullSimplify[ Integrate[ Boole[a < Norm[{x1, x2, x3}] < b]/ Norm[{x1, x2, x3}]^3, {x1, \(-b\), b}, {x2, \(-b\), b}, {x3, \(-b\), b}, Assumptions \[Rule] {0 < a < b}] \[Equal] 4 \[Pi]\ Log[b/a], {0 < a < b}]\ *) \)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.20", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is unable to evaluate the integral using the function ", StyleBox["Boole", FontFamily->"Courier New"], ". See also the file Ex6.20.pdf" }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(f[y1_, y2_, y3_] = Integrate[ Cos[{x1, x2, x3} . {y1, y2, y3}] Boole[x1^2 + x2^2 + x3^2 \[LessEqual] 1], {x1, \(-1\), 1}, {x2, \(-1\), 1}, {x3, \(-1\), 1}, Assumptions -> {Im[y1] == 0, Im[y2] \[Equal] 0, Im[y3] \[Equal] 0}];\)\), "\[IndentingNewLine]", \(g[y_List] = 4 \[Pi]/Norm[y]^2 \((Sin[Norm[y]]/Norm[y] - Cos[Norm[y]])\)\), "\[IndentingNewLine]", \(FullSimplify[ f[y1, y2, y3] == g[{y1, y2, y3}], {Im[y1] == 0, Im[y2] \[Equal] 0, Im[y3] \[Equal] 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.21", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is unable to evaluate the integral." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[\[ExponentialE]^\((x1^2 + x2^2 - x1^2 x2^2)\) Sqrt[1 - x1^2], {x1, 0, 1}, {x2, 0, 1}] \[Equal] \[Pi] \((\[ExponentialE] - 1)\)/4] /. x1 \[Rule] x\_1\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.23 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex6.23.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.36 (Sard's Theorem)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\n", \(R[a_] := {{Cos[a], Sin[a]}, {\(-Sin[a]\), Cos[a]}}\), "\n", \(arrow[x_, y_, h_, b_, a_] := Polygon[{R[a] . {\(-2\)\ b, \(-4\)\ h} + {x, y}, {x, y}, R[a] . {2\ b, \(-4\)\ h} + {x, y}, R[a] . {0, \(-3\)\ h} + {x, y}}]\), "\n", \(\(d = 1.6;\)\), "\n", \(\(a = 2/3;\)\), "\n", \(\(x0 = Sqrt[4/\((1 + a^2)\)];\)\), "\n", \(y[x_] := a\ x\), "\n", \(y1[x_] := a\ x + d\), "\n", \(y2[x_] := a\ x - d\), "\n", \(y3[x_] := \(-1\)/a\ x + x0\ \((a + 1/a)\)\), "\n", \(y4[x_] := \(-1\)/a\ x - x0\ \((a + 1/a)\)\), "\n", \(y5[x_] := a\ x - 2\ d\), "\n", \(y6[x_] := \(-1\)/a\ x - x0\ \((a + 1/a)\) - d\), "\n", \(\(p13 = {\((x0\ \((a + 1/a)\) - d)\)/\((a + 1/a)\), y1[\((x0\ \((a + 1/a)\) - d)\)/\((a + 1/a)\)]};\)\), "\n", \(\(p23 = {\((x0\ \((a + 1/a)\) + d)\)/\((a + 1/a)\), y2[\((x0\ \((a + 1/a)\) + d)\)/\((a + 1/a)\)]};\)\), "\n", \(\(p14 = {\((\(-x0\)\ \((a + 1/a)\) - d)\)/\((a + 1/a)\), y1[\((\(-x0\)\ \((a + 1/a)\) - d)\)/\((a + 1/a)\)]};\)\), "\n", \(\(p24 = {\((\(-x0\)\ \((a + 1/a)\) + d)\)/\((a + 1/a)\), y2[\((\(-x0\)\ \((a + 1/a)\) + d)\)/\((a + 1/a)\)]};\)\), "\n", \(\(Graphics[{Text["\", {0, \(-0.2\)}], Text["\", {3, y[2.8]}], Text["\", {\(-2.75\), y[\(-2.75\)]}], Text["\", {1.5, \(-2.5\)}], {AbsolutePointSize[4], Point[{0, 0}]}, {AbsoluteThickness[1], Circle[{0, 0}, 2], Line[{{\(-2.5\), y[\(-2.5\)]}, {3, y[3]}}], Line[{{\(-2.95\), y[\(-2.95\)]}, {\(-3.25\), y[\(-3.25\)]}}], Line[{{\(-3.1\), y6[\(-3.1\)]}, {\(-1.7\), y6[\(-1.7\)]}}], Line[{{\(-0.125\), y5[\(-0.125\)]}, {3.05, y5[3.05]}}], {Dashing[{0.02, 0.02}], Line[{{\(-4\), y1[\(-4\)]}, {2.5, y1[2.5]}}], Line[{{\(-2.5\), y2[\(-2.5\)]}, {4, y2[4]}}]}, {Dashing[{0.01, 0.01}], Line[{p23, {3.3, y3[3.3]}}], Line[{p24, {\(-0.025\), y4[\(-0.025\)]}}]}}, {AbsoluteThickness[ 1.75], Line[{p13, p23, p24, p14, p13}]}, arrow[\(-3.1\), y6[\(-3.1\)], 0.075, 0.06, \(-0.185\)\ \[Pi]], arrow[\(-1.7\), y6[\(-1.7\)], 0.075, 0.06, 0.81\ \[Pi]], arrow[\(-0.125\), y5[\(-0.125\)], 0.075, 0.06, 1.31\ \[Pi]], arrow[3.05, y5[3.05], 0.075, 0.06, 0.31\ \[Pi]]}, AspectRatio \[Rule] Automatic];\)\), "\n", \(\(Show[%, DefaultColor \[Rule] darkblue, Background \[Rule] lightblue, DefaultFont \[Rule] 12. , ImageSize \[Rule] 500];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.39 (zeta function and dilogarithm) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Sum[1/\((2 n + 1)\)^2, {n, 0, \[Infinity]}]\), "\[IndentingNewLine]", \(Integrate[ 1/\((1 - x1^2 x2^2)\), {x1, 0, 1}, {x2, 0, 1}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Zeta[2]\), "\[IndentingNewLine]", \(N[Zeta[2], 20]\), "\[IndentingNewLine]", \(N[Zeta[2] - Sum[1/n^2, {n, 1, 10000}], 12]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ 1/x\ Log[\((1 + x)\)/\((1 - x)\)], {x, 0, 1}]\), "\[IndentingNewLine]", \(f[x_] = Integrate[1/\((1 - x1^2 x2^2)\), {x1, 0, 1}, Assumptions \[Rule] {0 < x2 < 1}] /. x2 \[Rule] x\), "\[IndentingNewLine]", \(Integrate[ Normal[Series[f[xx], {xx, 0, 20}]], {xx, 0, x}]\), "\[IndentingNewLine]", \(Integrate[ Normal[Series[Log[1 - xx]/xx, {xx, 0, 20}]], {xx, 0, x}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Sum[\((\(-1\))\)^n/\((2 n + 1)\)^2, {n, 0, \[Infinity]}]\), "\[IndentingNewLine]", \(Integrate[ 1/\((1 + x1^2 x2^2)\), {x1, 0, 1}, {x2, 0, 1}]\), "\[IndentingNewLine]", \(Integrate[ArcTan[x]/x, {x, 0, 1}]\), "\[IndentingNewLine]", \(N[%]\), "\[IndentingNewLine]", \(Integrate[Log[Sin[t]], {t, 0, \[Pi]/2}]\), "\[IndentingNewLine]", \(Integrate[\(-Log[Abs[2 Sin[t/2]]]\), {t, 0, \[Pi]/2}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ 1/\((1 - x1^4 x2^4)\), {x1, 0, 1}, {x2, 0, 1}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ 1/\((1 - x1\ x2)\), {x1, 0, 1}, {x2, 0, 1}]\), "\[IndentingNewLine]", \(Integrate[ Log[x1\ x2]/\((1 - x1\ x2)\), {x1, 0, 1}, {x2, 0, 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.40", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["In part (ii)", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", FontSlant->"Italic"], " is unable to produce the standard expressions for \[Zeta](2", StyleBox["n", FontSlant->"Italic"], ") in terms of the Bernoulli numbers, neither is it able to give the \ standard evaluation in terms of Euler numbers for the series. " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[FullSimplify::fas]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(b[n_] := \((\(-1\))\)^\((n - 1)\)\ 1/2\ \((2 \[Pi])\)^\((2 n)\) BernoulliB[2 n]/\(\((2 n)\)!\)\), "\[IndentingNewLine]", \(FullSimplify[ Zeta[2 n] - b[n], {Element[n, Integers], n > 0}]\), "\[IndentingNewLine]", \(Table[b[n], {n, 1, 10}]\), "\[IndentingNewLine]", \(c[n_] := FullSimplify[ Zeta[2 n] - b[n], {Element[n, Integers], n > 0}]\), "\[IndentingNewLine]", \(Table[c[n], {n, 1, 10}]\), "\[IndentingNewLine]", \(d[n_] := FullSimplify[ Sum[\((\(-1\))\)^k/\((2 k + 1)\)^\((2 n + 1)\), {k, 0, \[Infinity]}], {Element[n, Integers], n > 0}]\), "\[IndentingNewLine]", \(d[n]\), "\[IndentingNewLine]", \(Table[\((\(-1\))\)^n\ EulerE[2 n], {n, 1, 10}]\), "\[IndentingNewLine]", \(e[n_] := 1/2 \((\(-1\))\)^n \((\[Pi]/2)\)^\((2 n + 1)\) EulerE[2 n]/\(\((2 n)\)!\)\), "\[IndentingNewLine]", \(Table[e[n], {n, 0, 10}]\), "\[IndentingNewLine]", \(f[n_] := FullSimplify[ d[n] - e[n], {Element[n, Integers], n > \(-1\)}]\), "\[IndentingNewLine]", \(Table[f[n], {n, 0, 9}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.43 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(f[x_] := 1/\((\[Sigma]\ Sqrt[ 2 \[Pi]])\) \[ExponentialE]^\((\(-\((x - \[Mu])\)^2\)/\((2\ \ \[Sigma]^2)\))\)\), "\[IndentingNewLine]", \(Integrate[f[x], {x, \(-\[Infinity]\), \[Infinity]}, Assumptions \[Rule] {\[Sigma] > 0}]\), "\[IndentingNewLine]", \(Integrate[\((x - \[Mu])\) f[x], {x, \(-\[Infinity]\), \[Infinity]}, Assumptions \[Rule] {\[Sigma] > 0}]\), "\[IndentingNewLine]", \(Integrate[\((x - \[Mu])\)^2 f[x], {x, \(-\[Infinity]\), \[Infinity]}, Assumptions \[Rule] {\[Sigma] > 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.49", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[StyleBox["See the file Ex6.49.pdf.", FontVariations->{"CompatibilityType"->0}]], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.50 (Euler's Gamma and Beta functions) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is unable to recognize the Beta function in part (ii) and cannot simplify \ the trigonometric integral in part (vii). ", StyleBox["See also the file Ex6.50.pdf.", FontVariations->{"CompatibilityType"->0}] }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(\((Clear["\<`*\>"]\[IndentingNewLine]Off[N::meprec]\[IndentingNewLine]\(\ lightblue = RGBColor[220/256, 248/256, 248/256];\))\)\^\[Placeholder]\), "\ \[IndentingNewLine]", \(\(hoek = 0.03 {1, 1};\)\), "\n", \(\(p1 = Plot[Gamma[p], {p, 0.025, 5}, PlotPoints \[Rule] 200, Ticks \[Rule] {{1, 2, 3, 4, 5}, {5, 15, 25}}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] {{0, 6}, {0, 30}}, PlotRegion \[Rule] {{0.15, 0.9}, {0.15, 0.9}}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p2 = Plot3D[Beta[p1, p2], {p1, hoek[\([1]\)], 1}, {p2, hoek[\([2]\)], 1}, PlotPoints \[Rule] {40, 30}, ViewPoint \[Rule] {1.5, 2.5, 0.7}, Axes \[Rule] None, PlotRange \[Rule] {0, Beta @@ hoek}, PlotRegion \[Rule] {{0, 1}, {0, 1}}, Boxed \[Rule] False, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p3 = ParametricPlot3D[{0, 0, t}, {t, 0, 95}, DefaultColor \[Rule] RGBColor[0, 0.5, 0], PlotPoints \[Rule] 100, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p4 = Graphics3D[{{AbsoluteThickness[1], RGBColor[0, 0.5, 0], Line[{{1.25, 0, 0}, {0, 0, 0}, {0, 1.5, 0}}]}, {AbsolutePointSize[4], Point[{1, 0, 0}], Point[{0, 1, 0}], Point[{0, 0, 75}]}, Text["\", {1, 0, 0}], Text["\", {0, 1, 0}], Text["\<75\>", {0, 0, 75}]}];\)\), "\n", \(\(p5 = Show[p2, p3, p4, PlotRange \[Rule] {0, 95}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[GraphicsArray[{p1, p5}], Background \[Rule] lightblue, ImageSize \[Rule] 1000, DisplayFunction \[Rule] $DisplayFunction];\)\[IndentingNewLine]\), "\ \[IndentingNewLine]", \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[N::meprec]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[\[ExponentialE]^\((\(-t\))\) t^\((p - 1)\), {t, 0, \[Infinity]}, Assumptions \[Rule] {Re[p] > 0}]\), "\[IndentingNewLine]", \(FullSimplify[ Gamma[n + 1], {n > 0, Element[n, Integers]}]\), "\[IndentingNewLine]", \(Integrate[ 2 \[ExponentialE]^\((\(-u^2\))\) u^\((2 p - 1)\), {u, 0, \[Infinity]}, Assumptions \[Rule] {Re[p] > 0}]\), "\[IndentingNewLine]", \(Gamma[1/2]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[ 2 Cos[\[Alpha]]^\((2 p\_1 - 1)\) Sin[\[Alpha]]^\((2 p\_2 - 1)\), {\[Alpha], 0, \[Pi]/2}] \[Equal] Beta[p\_1, p\_2]\ , {p\_1 > 0, p\_2 > 0}]\), "\[IndentingNewLine]", \(Table[ Integrate[ 2 Cos[\[Alpha]]^\((2 p1 - 1)\) Sin[\[Alpha]]^\((2 p2 - 1)\), {\[Alpha], 0, \[Pi]/2}] - Beta[p1, p2]\ , {p1, 1, 5}, {p2, 1, 5}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ Cos[\[Alpha]]^\((p\_1)\) Sin[\[Alpha]]^\((p\_2)\), {\[Alpha], 0, \[Pi]/2}, Assumptions \[Rule] {p\_1 > 0, p\_2 > 0}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[t^\((p\_1 - 1)\) \((1 - t)\)^\((p\_2 - 1)\), {t, 0, 1}, Assumptions \[Rule] {p\_1 > 0, p\_2 > 0}]\), "\[IndentingNewLine]", \(Integrate[ u^\((p\_1 - 1)\)/\((1 + u)\)^\((p\_1 + p\_2)\), {u, 0, \[Infinity]}, Assumptions \[Rule] {p\_1 > 0, p\_2 > 0}]\), "\[IndentingNewLine]", \(Integrate[ 2 v^\((2 p\_1 - 1)\)/\((1 + v^2)\)^\((p\_1 + p\_2)\), {v, 0, \[Infinity]}, Assumptions \[Rule] {p\_1 > 0, p\_2 > 0}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[u^\((m - 1)\)/Sqrt[1 - u^n], {u, 0, 1}, Assumptions \[Rule] {Re[n] > 0, Re[m] > 0}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[Beta[p, p] \[Equal] 2^\((1 - 2 p)\) Beta[p, 1/2], p > 0]\), "\[IndentingNewLine]", \(FullSimplify[ Gamma[2 p] \[Equal] 2^\((2 p - 1)\) \[Pi]^\((\(-1\)/2)\) Gamma[p] Gamma[p + 1/2], p > 0]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[1/Sqrt[1 - t^4], {t, 0, 1}] \[Equal] 1/\((4 Sqrt[2 \[Pi]])\) Gamma[1/4]^2]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ Gamma[n + 1/2] \[Equal] \(\((2 n)\)!\) Sqrt[\[Pi]]/\((2^\((2 n)\) \(n!\))\), {n \[GreaterEqual] 0, Element[n, Integers]}]\), "\[IndentingNewLine]", \(FullSimplify[ Sum[\((\(-1\))\)^k/\((2 k + 1)\)\ \ Binomial[n, k], {k, 0, n}] \[Equal] \((\(n!\))\)^2\ 2^\((2 n)\)/\(\((2 n + 1)\)!\), {n > 0, Element[n, Integers]}]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[\((1 - u^2)\)^n, {u, 0, 1}, Assumptions -> {n > 0, Element[n, Integers]}] \[Equal] \((\(n!\))\)^2\ 2^\((2 n)\)/\(\((2 n + 1)\)!\), {n > 0, Element[n, Integers]}]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[Cos[\[Alpha]]^\((2 n)\), {\[Alpha], 0, \[Pi]}, Assumptions \[Rule] {n > 0, Element[n, Integers]}] \[Equal] \(\((2 n - 1)\)!!\)/\(\((2 n)\)!!\) \[Pi], {n > 0, Element[n, Integers]}]\), "\[IndentingNewLine]", \(Table[ Integrate[Cos[\[Alpha]]^\((2 n)\), {\[Alpha], 0, \[Pi]}, Assumptions \[Rule] {n > 0, Element[n, Integers]}] - \(\((2 n - 1)\)!!\)/\(\((2 n)\)!!\) \[Pi], {n, 1, 10}]\), "\[IndentingNewLine]", \(Normal[Series[ArcSin[x], {x, 0, 21}]]\), "\[IndentingNewLine]", \(x + Sum[\(\((2 n - 1)\)!!\)/\(\((2 n)\)!!\)\ x^\((2 n + 1)\)/\((2 n + 1)\), {n, 1, 10}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[\[Pi]^\((k)\)/Gamma[k + 1], {k > 0, Element[k, Integers]}]\), "\[IndentingNewLine]", \(FullSimplify[\[Pi]^\((\((2 k - 1)\)/2)\)/ Gamma[\((2 k - 1)\)/2 + 1] \[Equal] 2^\((2 k)\) \[Pi]^\((k - 1)\) \(k!\)/\(\((2 k)\)!\), {k > 0, Element[k, Integers]}]\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(t = Table[\[Pi]^\((n/2)\)/Gamma[n/2 + 1], {n, 1, 30}]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(ListPlot[t, PlotJoined \[Rule] True, TextStyle \[Rule] {FontFamily -> "\", FontSize \[Rule] 10. }, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.52", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["In part (ii)", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", FontSlant->"Italic"], " is unable to produce the standard expressions for \[Zeta](2", StyleBox["n", FontSlant->"Italic"], ") in terms of the Bernoulli numbers, neither is it able to evaluate the \ integral in part (iii) or to give the standard evaluation in terms of Euler \ or Bernoulli numbers for the integrals in part (iv). " }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(f[s_] := Integrate[ x^\((s - 1)\)/\((\[ExponentialE]^x - 1)\), {x, 0, \[Infinity]}, Assumptions \[Rule] {Re[s] > 1}]\), "\[IndentingNewLine]", \(f[s]\), "\[IndentingNewLine]", \(FullSimplify[PolyLog[s, 1] \[Equal] Zeta[s], Assumptions \[Rule] {Re[s] > 1}]\), "\[IndentingNewLine]", \(\(\(Integrate[ x^\((s - 1)\)/\((\[ExponentialE]^x + 1)\), {x, 0, \[Infinity]}, Assumptions \[Rule] {Re[s] > 1}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(f[2 n]\), "\[IndentingNewLine]", \(Integrate[Log[x]^\((2 n - 1)\)/\((x - 1)\), {x, 0, 1}, Assumptions -> {Element[n, Integers], n > 0}]\), "\[IndentingNewLine]", \(g[n_] := FullSimplify[ Zeta[2 n] - \((\(-1\))\)^\((n - 1)\) 1/2\ \((2 \[Pi])\)^\((2 n)\) BernoulliB[2 n]/\(\((2 n)\)!\), {Element[n, Integers], n > 0}]\), "\[IndentingNewLine]", \(g[n]\), "\[IndentingNewLine]", \(Table[g[n], {n, 1, 10}]\), "\[IndentingNewLine]", \(\(\(FullSimplify[ Integrate[ x^\((2 n - 1)\)/\((\[ExponentialE]^x - 1)\), {x, 0, \[Infinity]}] - \((\(-1\))\)^\((n - 1)\) \((2 \[Pi])\)^\((2 n)\) BernoulliB[2 n]/\((4 n)\), {Element[n, Integers], n > 0}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ x^s\ \[ExponentialE]^x/\((\[ExponentialE]^x - 1)\)^2, {x, 0, \[Infinity]}]\), "\[IndentingNewLine]", \(\(\(FullSimplify[ Integrate[ x^s\ \[ExponentialE]^x/\((\[ExponentialE]^x - 1)\)^2, {x, 0, \[Infinity]}] \[Equal] Gamma[s + 1] Zeta[s], Re[s] > 1]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[x^\((2 n)\)/Cosh[x], {x, 0, \[Infinity]}, Assumptions -> {Element[n, Integers], n > 0}]\), "\[IndentingNewLine]", \(Integrate[2 Log[x]^\((2 n)\)/\((x^2 + 1)\), {x, 0, 1}, Assumptions -> {Element[n, Integers], n > 0}]\), "\[IndentingNewLine]", \(h[n_] := FullSimplify[ Integrate[ x^\((2 n)\)/Cosh[x], {x, 0, \[Infinity]}] - \((\(-1\))\)^ n \((\[Pi]/2)\)^\((2 n + 1)\) EulerE[2 n], {Element[n, Integers], n > \(-1\)}]\), "\[IndentingNewLine]", \(Table[\((\(-1\))\)^n\ EulerE[2 n], {n, 1, 5}]\), "\[IndentingNewLine]", \(Table[h[n], {n, 1, 10}]\), "\[IndentingNewLine]", \(i[n_] := Integrate[x^\((2 n - 1)\)/Sinh[x], {x, 0, \[Infinity]}, Assumptions -> {Element[n, Integers], n > 0}]\), "\[IndentingNewLine]", \(i[n]\), "\[IndentingNewLine]", \(k[n_] := FullSimplify[ i[n] - \((\(-1\))\)^\((n - 1)\) \((2^\((2 n)\) - 1)\) \[Pi]^\((2 n)\) BernoulliB[2 n]/\((2 n)\), {Element[ n, Integers], n > 0}]\), "\[IndentingNewLine]", \(k[n]\), "\[IndentingNewLine]", \(\(\(Table[k[n], {n, 1, 10}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ x^3/\((\[ExponentialE]^x - 1)\), {x, 0, \[Infinity]}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.57", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["\<\ Clear[\"`*\"] lightblue=RGBColor[220/256,248/256,248/256]; ParametricPlot3D[Append[{x,y,Abs[Gamma[x+I \ y]]},SurfaceColor[RGBColor[0.7,0.7,1]]]//Evaluate,{x, -3.5, 3.2}, {y, -2, \ 2.2}, LightSources\[Rule]{{{-3,-1,0},RGBColor[1,0,0]},{{-3,-1,0},RGBColor[1,0,0]},{{\ -3,-1,0},RGBColor[1,0,0]},{{0,1,0},RGBColor[1,1,0]},{{0,2,2},RGBColor[0,1,1]}}\ ,Background\[Rule]lightblue, PlotRange->{-1,10},PlotPoints -> {151,110}, \ BoxRatios->Automatic,ViewPoint->{-1,-2,1},Boxed->False,Axes->None,ImageSize->\ 700];\ \>", "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.58 (Reflection formula for Gamma function) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Appearances deceive, the plots look more alarmingly then they \ actually are", FontVariations->{"CompatibilityType"->0}], StyleBox[". ", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}] }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[\[Infinity]::indet]\), "\[IndentingNewLine]", \(Off[N::meprec]\), "\[IndentingNewLine]", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(f[x_] := Gamma[x] Gamma[1 - x] Sin[\[Pi]\ x]\), "\[IndentingNewLine]", \(ff[x_] := Gamma[1 + x] Gamma[1 - x] Sin[\[Pi]\ x]/x\), "\[IndentingNewLine]", \(FullSimplify[f[x] \[Equal] ff[x]]\), "\[IndentingNewLine]", \(FullSimplify[ Table[D[f[x], {x, j}], {j, 0, 5}]]\), "\[IndentingNewLine]", \(Table[D[f[x], {x, j}], {j, 0, 5}] /. x \[Rule] 0\), "\[IndentingNewLine]", \(Table[ Limit[D[f[x], {x, j}], x \[Rule] 0], {j, 0, 5}]\), "\[IndentingNewLine]", \(\(Plot[f[x], {x, 1 - 0.00001, 1 + 0.00001}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, PlotPoints \[Rule] 300, ImageSize \[Rule] 800];\)\), "\[IndentingNewLine]", \(\(Plot[f[x], {x, 1 - 0.00001, 1 + 0.00001}, PlotRange \[Rule] {\[Pi] - 0.01, \[Pi] + 0.01}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, PlotPoints \[Rule] 300, ImageSize \[Rule] 800];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[f[x + 1] \[Equal] f[x]]\), "\[IndentingNewLine]", \(\(Plot[f[x], {x, 0, 10}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotPoints \[Rule] 300, ImageSize \[Rule] 800];\)\), "\[IndentingNewLine]", \(\(\(Plot[f[x], {x, 0, 10}, PlotRange \[Rule] {\[Pi] - 0.01, \[Pi] + 0.01}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotPoints \[Rule] 300, ImageSize \[Rule] 800];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(\(FullSimplify[ f[x/2] f[\((x + 1)\)/2] \[Equal] \[Pi]\ f[x]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(g[x_] := D[Log[f[xx]], {xx, 2}] /. xx \[Rule] x\), "\[IndentingNewLine]", \(\(Plot[g[x], {x, 0, 10}, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, PlotPoints \[Rule] 300, ImageSize \[Rule] 800];\)\), "\[IndentingNewLine]", \(FullSimplify[ g[x/2] + g[\((x + 1)\)/2] \[Equal] \[Pi]\ 4 g[x]]\), "\[IndentingNewLine]", \(\(\(FullSimplify[Gamma[x] Gamma[1 - x]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(\(Integrate[1/\((1 + x^n)\), {x, 0, \[Infinity]}, Assumptions \[Rule] {Element[n, Integers], n > 1}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[1/Sqrt[1 - t^3], {t, 0, 1}]\), "\[IndentingNewLine]", \(FullSimplify[% \[Equal] 1/\((2 \[Pi]\ 3^\((1/2)\) 2^\((1/3)\))\) Gamma[1/3]^3]\), "\[IndentingNewLine]", \(N[%%, 20]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 6.59 (Another proof of reflection formula for Gamma \ function) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[StyleBox["See the file Ex6.59.pdf.", FontVariations->{"CompatibilityType"->0}]], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.60 (Fresnel's integral) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(f[p_] = Integrate[Sin[x]/x^p, {x, 0, \[Infinity]}, Assumptions \[Rule] {0 < p < 2}]\), "\[IndentingNewLine]", \(FullSimplify[f[p] == \[Pi]/\((2 Gamma[p] Sin[p\ \[Pi]/2])\), 0 < p < 1]\), "\[IndentingNewLine]", \(Integrate[ Sin[x]/Sqrt[x], {x, 0, \[Infinity]}]\), "\[IndentingNewLine]", \(Integrate[ Sin[x]/\((x\ Sqrt[x])\), {x, 0, \[Infinity]}]\), "\[IndentingNewLine]", \(Integrate[Sin[x^2], {x, 0, \[Infinity]}]\), "\[IndentingNewLine]", \(Integrate[Cos[x]/x^p, {x, 0, \[Infinity]}, Assumptions \[Rule] {0 < p < 1}]\), "\[IndentingNewLine]", \(FullSimplify[% == \[Pi]/\((2 Gamma[p] Cos[p\ \[Pi]/2])\), 0 < p < 1]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[ 1/Gamma[p] t^\((p - 1)\) \[ExponentialE]^\((\(-x\)\ t)\), {t, 0, \[Infinity]}, Assumptions \[Rule] {x > 0, 0 < p < 1}]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(1/Gamma[p] Integrate[t^\((p - 1)\)/\((1 + t^2)\), {t, 0, \[Infinity]}, Assumptions \[Rule] {0 < p < 1}]\), "\[IndentingNewLine]", \(FullSimplify[f[p] \[Equal] %, 0 < p < 1]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[x^\((s - 1)\) Sin[x], {x, 0, \[Infinity]}, Assumptions \[Rule] {\(-1\) < s < 1}]\), "\[IndentingNewLine]", \(Integrate[x^\((s - 1)\) Cos[x], {x, 0, \[Infinity]}, Assumptions \[Rule] {0 < s < 1}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.69", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["As is not unusual,", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", FontSlant->"Italic"], " has problems to recognize standard definitions, which occur in its \ documentation. This is the case in part (vi)." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(Gamma[ c]\/\(Gamma[b] Gamma[c - b]\)\) Integrate[\(t\^\(b - 1\)\) \(\((1 - t)\)\^\(c - b - 1\)\) \((1 - z\ t)\)\^\(-a\), {t, 0, 1}, Assumptions -> {Im[a] \[Equal] Im[b] \[Equal] Im[c] \[Equal] 0, 0 < b < c, Re[z] < 1}]\), "\[IndentingNewLine]", RowBox[{\(FullSimplify[\(Gamma[c]\/\(Gamma[b] Gamma[c - b]\)\) Integrate[\(t\^\(b - 1\)\) \(\((1 - t)\)\^\(c - b - 1\)\) \((1 - z\ t)\)\^\(-a\), {t, 0, 1}, Assumptions -> {Im[a] \[Equal] Im[b] \[Equal] Im[c] \[Equal] 0, 0 < b < c, Re[z] < 1}] \[Equal] Hypergeometric2F1[a, b, c, z]]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(Simplify[ Pochhammer[a, n] \[Equal] Gamma[a + n]\/Gamma[a]]\), "\[IndentingNewLine]", \(FullSimplify[ Pochhammer[a, n] \[Equal] Gamma[a + n]\/Gamma[a]]\), "\[IndentingNewLine]", RowBox[{\(\[Sum]\+\(n = 0\)\%\[Infinity]\( Pochhammer[a, n]\/\(n!\)\) \((zt)\)\^n\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(k = 5;\), "\[IndentingNewLine]", \(\(hypergeo[a_, b_, c_]\)[z_] := Hypergeometric2F1[a, b, c, z]\), "\[IndentingNewLine]", \(Normal[ Series[\(hypergeo[a, b, c]\)[z], {z, 0, k}]]\), "\[IndentingNewLine]", \(\(hg[k_]\)[ z_] := \[Sum]\+\(n = 0\)\%k\(\( Pochhammer[a, n] Pochhammer[b, n]\)\/\(Pochhammer[c, n] \(n!\)\)\) z\^n\), "\[IndentingNewLine]", \(\(hg[k]\)[ z]\), "\[IndentingNewLine]", RowBox[{\(\(hg[\[Infinity]]\)[z]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(hypergeo[1, 1, 1]\)[ z]\), "\[IndentingNewLine]", \(\(hypergeo[\(-n\), 1, 1]\)[\(-z\)]\), "\[IndentingNewLine]", \(z\ \(hypergeo[1, 1, 2]\)[\(-z\)]\), "\[IndentingNewLine]", \(z\ \(hypergeo[1\/2, 1\/2, 3\/2]\)[z\^2]\), "\[IndentingNewLine]", RowBox[{\(z\ \(hypergeo[1\/2, 1, 3\/2]\)[\(-z\^2\)]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\ \), "\[IndentingNewLine]", \(Clear[ k]\), "\[IndentingNewLine]", \(d[f_, a_, z] := z\ \[PartialD]\_z\ f[z] + a\ f[z]\), "\[IndentingNewLine]", \(\(\(dhg[k_]\)[a_]\)[z_] := d[hg[k], a, z]\), "\[IndentingNewLine]", \(k = 5;\), "\[IndentingNewLine]", \(Collect[\(\(dhg[k]\)[a]\)[z], z]\), "\[IndentingNewLine]", \(a[ j_] := \(Collect[z\ d[\(dhg[k]\)[b], a, z], z]\)[\([j]\)]\), "\[IndentingNewLine]", \(b[ j_] := \(Collect[d[\(dhg[k]\)[c - 1], 0, z], z]\)[\([j]\)]\), "\[IndentingNewLine]", \(Table[ Factor[a[j]], {j, 1, k}]\), "\[IndentingNewLine]", RowBox[{\(Table[Factor[a[j]], {j, 1, k}]\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(hgdiff[f_, z_] := Collect[FullSimplify[ z \((1 - z)\) D[f[z], {z, 2}] + \((c - \((a + b + 1)\) z)\) D[f[z], z] - a\ b\ f[z]], z]\), "\[IndentingNewLine]", RowBox[{\(Simplify[hgdiff[hg[k], z]]\), " "}], "\[IndentingNewLine]", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"hgdiff", "[", RowBox[{\(hypergeo[a, b, c]\), StyleBox[",", "MR"], StyleBox["z", "MR"]}], StyleBox["]", "MR"]}], StyleBox["==", "MR"], StyleBox["0", "MR"]}], StyleBox["]", "MR"]}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 6.83 (Another proof of Example 6.11.4)", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(1/2 FourierTransform[\[ExponentialE]\^\(\(-\(1\/2\)\)\ x\^2\), x, \[Xi], FourierParameters \[Rule] {1, \(-1\)}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 6.91 (Partial-fraction decomposition of hyperbolic \ functions)\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(\(FourierTransform[\[ExponentialE]\^\(\(-t\)\ Abs[x]\), x, \[Xi], FourierParameters \[Rule] {1, \(-1\)}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[ Sin[\[Xi]\ x]\/\(\[ExponentialE]\^x - 1\)\ , {x, 0, \[Infinity]}], Re[\[Xi]] > 0]\), "\[IndentingNewLine]", \(FullSimplify[Integrate[Sin[\[Xi]\ x]\/Sinh[x], {x, 0, \[Infinity]}], Re[\[Xi]] > 0]\), "\[IndentingNewLine]", \(FullSimplify[Integrate[Cos[\[Xi]\ x]\/Cosh[x], {x, 0, \[Infinity]}], Re[\[Xi]] > 0]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 7: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 7.2", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is unable to perform the direct calculation." }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\n", \(Print["\"]\), "\n", \(FullSimplify[Integrate[Norm[{1, \(Log'\)[t]}], {t, 1, x}], x > 0]\), "\n", \(Print["\"]\), "\n", \(f[t_] := Log[t]\), "\n", \(\[Phi][t_] := {t, f[t]}\), "\n", \(i[t_] := Simplify[Norm[\(\[Phi]'\)[tt]], tt > 0] /. tt \[Rule] t\), "\n", \(int[t_] := Simplify[Integrate[i[tt], tt], tt > 0] /. tt \[Rule] t\), "\n", \(int[x] - int[1]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Exercise 7.5 (Lemniscate, \[CapitalGamma](", Cell[BoxData[ \(TraditionalForm\`1\/4\)]], ") and elliptic integrals of the first kind) " }], "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["'s definition of the complete elliptic integral of the first kind \ with modulus ", FontVariations->{"CompatibilityType"->0}], StyleBox["k ", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["is not the usual one, as customary in pure mathematics. In fact, \ ", FontVariations->{"CompatibilityType"->0}], StyleBox["K", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["(", FontVariations->{"CompatibilityType"->0}], StyleBox["k", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[") in the book is given by EllipticK[", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`k\^2\)]], StyleBox["] in ", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica.", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" Furthermore, it does not recognize the power series expansion \ of this elliptic integral. Finally, it is not very successful at comparing \ the values of definite integrals over real intervals with real integrands: it \ produces complex values. ", FontVariations->{"CompatibilityType"->0}] }], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[General::spell1]\), "\[IndentingNewLine]", \(Off[N::meprec]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(int[\[Phi]_] = Integrate[1\/\@Cos[2 \[Alpha]], {\[Alpha], \[Phi], \[Pi]\/4}, Assumptions \[Rule] {0 < \[Phi] < \[Pi]\/4}];\)\), "\ \[IndentingNewLine]", \(\(iint[\[Phi]_] = Integrate[1\/\@\(1 - t\^4\), {t, 0, \@Cos[2 \[Phi]]}, Assumptions \[Rule] {0 < \[Phi] < \[Pi]\/4}];\)\), "\ \[IndentingNewLine]", \(FullSimplify[int[\[Phi]] == iint[\[Phi]], Assumptions \[Rule] {0 < \[Phi] < \[Pi]\/4}]\), "\[IndentingNewLine]", \(Simplify[\[PartialD]\_\[Phi]\ \((int[\[Phi]] - iint[\[Phi]])\), Assumptions \[Rule] {0 < \[Phi] < \[Pi]\/4}]\), "\[IndentingNewLine]", \(\((int[\[Phi]] - iint[\[Phi]])\) /. \[Phi] \[Rule] \[Pi]\/4\), \ "\[IndentingNewLine]", \(FullSimplify[ int[0] == Gamma[1\/4]\^2\/\(4 \@\( 2 \[Pi]\)\)]\), "\[IndentingNewLine]", \(FullSimplify[ iint[0] == Gamma[1\/4]\^2\/\(4 \@\( 2 \[Pi]\)\)]\), "\[IndentingNewLine]", \(N[Integrate[1\/\@Cos[2 \[Alpha]], {\[Alpha], 0, \[Pi]\/4}] - Gamma[1\/4]\^2\/\(4 \@\( 2 \[Pi]\)\)]\), "\[IndentingNewLine]", \(NIntegrate[1\/\@Cos[2 \[Alpha]], {\[Alpha], 0, \[Pi]\/4}] - Gamma[1\/4]\^2\/\(4 \@\( 2 \[Pi]\)\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ EllipticK[1\/2] \[Equal] Gamma[1\/4]\^2\/\(4 \@ \[Pi]\)]\), "\[IndentingNewLine]", \(N[EllipticK[1\/2] - Gamma[1\/4]\^2\/\(4 \@ \[Pi]\)]\), "\[IndentingNewLine]", \(N[Gamma[1\/4], 20]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(\[Pi]\/2\) Hypergeometric2F1[1\/2, 1\/2, 1, k] == EllipticK[k]\), "\[IndentingNewLine]", \(FullSimplify[\(\[Pi]\/2\) \((1 + \[Sum]\+\(n = \ 1\)\%\[Infinity]\((\(\(\((2 n - 1)\)!!\)\/\(\((2 n)\)!!\)\) k\^n)\)\^2)\) == EllipticK[k\^2]]\), "\[IndentingNewLine]", \(Series[ Hypergeometric2F1[1\/2, 1\/2, 1, k\^2], {k, 0, 21}]\), "\[IndentingNewLine]", \(Series[\(2\/\[Pi]\) EllipticK[k\^2], {k, 0, 21}]\), "\[IndentingNewLine]", \(FullSimplify[ Gamma[1\/4]\^2 \[Equal] 2 \( \[Pi]\^\(3\/2\)\) Hypergeometric2F1[1\/2, 1\/2, 1, 1\/2]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 7.15 (Viviani's solid) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See also the file Ex7.15.pdf ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_] := {Cos[\[Alpha]]\^2, Cos[\[Alpha]] Sin[\[Alpha]], Sin[\[Alpha]]}\), "\n", \(\(\(ParametricPlot3D[{Append[\[Phi][\[Alpha]], RGBColor[1, 0, 0]], Append[{Cos[\[Alpha]]\^2, Cos[\[Alpha]] Sin[\[Alpha]], 0}, RGBColor[1, 0, 0]], Append[{Cos[\[Alpha]], 0, Sin[\[Alpha]]}, darkblue], Append[{Cos[\[Alpha]], Sin[\[Alpha]], 0}, darkblue], Append[{0, Cos[\[Alpha]], Sin[\[Alpha]]}, darkblue]} // Evaluate, {\[Alpha], 0, \[Pi]/2}, Boxed \[Rule] False, Ticks \[Rule] None, AxesEdge \[Rule] {{\(-1\), \(-1\)}, {\(-1\), \(-1\)}, {\(-1\), \ \(-1\)}}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] {{0, 1.5}, {0, 1.2}, {0, 1.15}}, ViewPoint \[Rule] {10, 3, 1}, Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\n", \(Integrate[r\ \@\(1 - r\^2\), r]\), "\n", \(Integrate[r\ \@\(1 - r\^2\), {r, 0, Cos[\[Alpha]]}, Assumptions \[Rule] {0 < \[Alpha] < \[Pi]\/2}]\), "\n", \(2\/3\ Integrate[ 1 - Abs[Sin[\[Alpha]]\^3], {\[Alpha], \(-\(\[Pi]\/2\)\), \ \[Pi]\/2}]\), "\n", \(Integrate[Sin[\[Alpha]]\^3, {\[Alpha], 0, \[Pi]\/2}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 7.16 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[ \(\(\(Clear["\<`*\>"]\[IndentingNewLine] \(lightblue = RGBColor[220/256, 248/256, 248/256];\)\[IndentingNewLine] \(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\[IndentingNewLine]\[IndentingNewLine] (*\ R = 1\ and\ x_j\ equals\ r\ *) \[IndentingNewLine] \(p1 = Plot[\(-\(r\/3\)\), {r, 0, 1}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine] \(p2 = Plot[\(-\(1\/\(3 r\^2\)\)\), {r, 1, 3}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine] \(b1 = Graphics[{lightgreen, Polygon[{{0, 0}, {0, \(-1\)/3 - 0.05}, {1, \(-1\)/3 - 0.05}, {1, 0}}]}];\)\[IndentingNewLine] \(Show[b1, p1, p2, PlotRange \[Rule] {\(-1\)/3 - 0.05, 0}, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction -> $DisplayFunction];\)\[IndentingNewLine]\ \[IndentingNewLine] (*\ norm\ of\ x\ equals\ r\ *) \[IndentingNewLine] \(q1 = Plot[\(1\/3\) \((r\^2 + 3)\), {r, 0, 1}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine] \(q2 = Plot[\(1\/\(3 r\)\) \((3 r\^2 + 1)\), {r, 1, 3}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine] \(b2 = Graphics[{lightgreen, Polygon[{{0, 0}, {0, 3.3}, {1, 3.3}, {1, 0}}]}];\)\[IndentingNewLine] \(Show[b2, q1, q2, PlotRange \[Rule] {0, 3.3}, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction -> $DisplayFunction];\)\)\(\ \)\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 7.17 (Unrolling a cone) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["\<\ See also the file Ex7.17.pdf. Here the representative example of a \ circular cone is given. \ \>", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[ General::spell1]\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\[IndentingNewLine]", \(view = {18, 3, 3};\), "\[IndentingNewLine]", \(\[Gamma][s_] := {Cos[s], Sin[s], k}\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(k = 1;\), "\[IndentingNewLine]", \(ParametricPlot3D[ Append[\[Gamma][s], RGBColor[1, 0, 0]] // Evaluate, {s, 0, 2 \[Pi]}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, Ticks \[Rule] None, AxesEdge \[Rule] {{\(-1\), \(-1\)}, {\(-1\), \(-1\)}, {\(-1\), \ \(-1\)}}, PlotPoints \[Rule] 20, ViewPoint \[Rule] view, Background \[Rule] lightblue, ImageSize \[Rule] 500];\), "\[IndentingNewLine]", \(Clear[ k]\), "\[IndentingNewLine]", \(cross[s_] := Simplify[Cross[\[Gamma][s], \(\[Gamma]'\)[ s]]]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Simplify[ cross[s] \[Equal] 0]\), "\[IndentingNewLine]", \(\[Phi][s_, t_] := t\ \[Gamma][ s]\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(k = 1;\), "\[IndentingNewLine]", \(ParametricPlot3D[ Append[\[Phi][s, t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, 0, 2 \[Pi]}, {t, 0, 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], PlotRange \[Rule] All, AspectRatio \[Rule] 1, Boxed \[Rule] False, Ticks \[Rule] None, Axes \[Rule] None, PlotPoints \[Rule] 150, ViewPoint \[Rule] view, Background \[Rule] lightblue, ImageSize \[Rule] 500];\), "\[IndentingNewLine]", \(Clear[ k]\), "\[IndentingNewLine]", \(normcross[s_] := Simplify[Norm[cross[s]], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(a1 = Integrate[1\/2\ normcross[s], {s, 0, 2 \[Pi]}, Assumptions \[Rule] {Im[k] \[Equal] 0}]\), "\[IndentingNewLine]", \(norm[s_] := Simplify[Norm[\[Gamma][s]], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(\[Alpha][s_] := Integrate[normcross[\[Sigma]]\/norm[\[Sigma]]\^2, {\[Sigma], 0, s}, Assumptions \[Rule] {s > 0, Im[k] \[Equal] 0}]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\[Alpha](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(\[Alpha][s]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\[CapitalUpsilon](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\),\!\(\* StyleBox[\"t\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(\[CapitalUpsilon][s_, t_] := Simplify[{t\ norm[s], \[Alpha][s]}, Im[k] \[Equal] 0]\), "\[IndentingNewLine]", \(\[CapitalUpsilon][s, t]\), "\[IndentingNewLine]", \(sol[a_] := Flatten[Solve[\[Alpha][ss] \[Equal] a, ss]]\), "\[IndentingNewLine]", \(\[Beta][a_] := ss /. sol[ a]\), "\[IndentingNewLine]", \(Print["\<\[Beta](\[Alpha])\>"]\), "\ \[IndentingNewLine]", \(\[Beta][\[Alpha]]\), "\[IndentingNewLine]", \(r[a_] := Simplify[norm[\[Beta][a]], Im[a] \[Equal] Im[k] == 0]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\!\(\* StyleBox[\"r\",\nFontSlant->\"Italic\"]\)(\[Alpha])\>\"", "]"}], "\[IndentingNewLine]", \(r[\[Alpha]]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\)))\>\"", "]"}], "\[IndentingNewLine]", \(a2 = FullSimplify[ Integrate[1\/2\ r[\[Alpha]]\^2, {\[Alpha], 0, \[Alpha][2 \[Pi]]}], Im[k] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[ a1 \[Equal] a2, k > 0]\), "\[IndentingNewLine]", \(\[Upsilon][s_, t_] := Simplify[t\ norm[s] {Cos[\[Alpha][s]], Sin[\[Alpha][s]]}, Im[k] \[Equal] 0]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\[Upsilon](\!\(\* StyleBox[\"t\",\nFontSlant->\"Italic\"]\)\[Gamma](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\)))\>\"", "]"}], "\[IndentingNewLine]", \(\[Upsilon][s, t]\), "\[IndentingNewLine]", \(k = 1;\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\[Upsilon](\!\(\* StyleBox[\"C\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(f[x_, y_] = \(\[Upsilon][x, y]\)[\([1]\)];\), "\n", \(g[x_, y_] = \(\[Upsilon][x, y]\)[\([2]\)];\), "\n", \(xmin = 0;\), "\n", \(xmax = 2 \[Pi];\), "\n", \(ymin = 0;\), "\n", \(ymax = 1;\), "\n", \(imax = 31;\), "\n", \(jmax = 31;\), "\n", \(plotpoints = 500;\), "\n", \(ParametricPlot[ Evaluate[ Join[Table[{f[xmin + t \((xmax - xmin)\), ymin + j \((ymax - ymin)\)/jmax], g[xmin + t \((xmax - xmin)\), ymin + j \((ymax - ymin)\)/jmax]}, {j, 0, jmax}], \[IndentingNewLine]Table[{f[ xmin + i \((xmax - xmin)\)/imax, ymin + t*\((ymax - ymin)\)], g[xmin + i \((xmax - xmin)\)/imax, ymin + t \((ymax - ymin)\)]}, {i, 0, imax}]]], {t, 0, 1}, AspectRatio \[Rule] Automatic, PlotRange \[Rule] All, PlotPoints \[Rule] plotpoints, Axes \[Rule] None, DefaultColor \[Rule] darkblue, Background \[Rule] lightblue, ImageSize \[Rule] 500];\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 7.19 (Pseudo-sphere and non-Euclidean geometry) \ \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Phi][s_] := {Sin[s], Cos[s] + Log[Tan[s\/2]]}\), "\[IndentingNewLine]", \(\(l = \@\(\[Sum]\+\(j = 1\)\%2\((\[PartialD]\_s\ \ \(\[Phi][s]\)[\([j]\)])\)\^2\);\)\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[ l, {s, t, \[Pi]\/2}], {0 < t < \[Pi]\/2}]\), "\[IndentingNewLine]", \(\(\(FullSimplify[ Integrate[l, {s, t, \[Pi]\/2}] \[Equal] Log[1\/Sin[t]], {0 < t < \[Pi]\/2}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Psi][s_, \[Alpha]_] := {Sin[s] Cos[\[Alpha]], Sin[s] Sin[\[Alpha]], Cos[s] + Log[Tan[s\/2]]}\), "\[IndentingNewLine]", \(c = FullSimplify[ Cross[\[PartialD]\_s\ \[Psi][ s, \[Alpha]], \[PartialD]\_\[Alpha]\ \[Psi][ s, \[Alpha]]]]\), "\[IndentingNewLine]", \(n = Simplify[\@\(\[Sum]\+\(j = 1\)\%3 c[\([j]\)]\^2\), Assumptions -> {Im[\[Alpha]] \[Equal] 0, 0 < s < \[Pi]}]\), "\[IndentingNewLine]", \(Integrate[n, {\[Alpha], \(-\[Pi]\), \[Pi]}, {s, 0, \[Pi]}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Exercise 7.21 (Spherical coordinates in ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["R", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], "n"], TraditionalForm]]], " and hyperarea of (", StyleBox["n", FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["-1", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], ")-sphere) " }], "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex7.21.pdf. ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[{\(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\\"Italic\"]\^\(n - 1\)\) is maximal if \!\(\* StyleBox[\"n\",\nFontSlant->\"Italic\"]\) = 7\>\"", "]"}], "\n", \(t = Table[\(2 \[Pi]\^\(n\/2\)\)\/Gamma[n\/2], {n, 1, 30}]\), "\[IndentingNewLine]", \(ListPlot[t, PlotJoined \[Rule] True, TextStyle \[Rule] {FontFamily -> "\", FontSize \[Rule] 10. }, PlotStyle \[Rule] RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, ImageSize \[Rule] 600];\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 7.44", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[StyleBox["See the file Ex7.44.pdf.", FontVariations->{"CompatibilityType"->0}]], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 7.46", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[StyleBox["See the file Ex7.46.pdf.", FontVariations->{"CompatibilityType"->0}]], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 7.53 (Spherical mean, Mean Value Theorem for a harmonic \ function, Darboux's equation, and Liouville's Theorem) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell["See the file Ex7.53.pdf. ", "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exercise 7.68 (Another computation of Newton's potential of a \ ball)\ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\), "\[IndentingNewLine]", \(\(\(Print[\*"\"\\""]\)\(\ \[IndentingNewLine]\) \) (*\ R = 1\ and\ vol\ A\ equals\ 4 \[Pi]/3\ *) \), "\[IndentingNewLine]", \(\(p1 = Plot[\(-\(1\/6\)\) \((3 - r\^2)\), {r, 0, 1}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = Plot[\(-\(1\/\(3 r\)\)\), {r, 1, 5}, PlotStyle -> RGBColor[1, 0, 0], AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(b = Graphics[{lightgreen, Polygon[{{0.01, \(- .01\)}, {0.01, \(-0.55\) - 0.05}, {1, \(-0.55\) - 0.05}, {1, \(-0.01\)}}]}];\)\), "\[IndentingNewLine]", \(\(Show[b, p1, p2, PlotRange \[Rule] {\(-0.55\), 0 + 0.05}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Axes \[Rule] True, Ticks \[Rule] None, AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, ImageSize \[Rule] 1000, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["CHAPTER 8: EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["Exercise 8.3 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\), "\[IndentingNewLine]", \(\[Phi]1[t_] := {t, t}\), "\[IndentingNewLine]", \(\[Phi]2[t_] := {t, t\^2}\), "\[IndentingNewLine]", \(Solve[\[Phi]1[t] == \[Phi]2[t], t] /. {Rule \[Rule] Equal}\), "\[IndentingNewLine]", \(\(p1 = ParametricPlot[\[Phi]1[t], {t, \(-0.1\), 1.1}, PlotStyle -> RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p2 = ParametricPlot[\[Phi]2[t], {t, \(-0.1\), 1.1}, PlotStyle -> RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p3 = ParametricPlot[\[Phi]2[t], {t, 0, 1. }, PlotStyle \[Rule] lightgreen, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(p4 = p3 /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\), "\[IndentingNewLine]", \(\(\(Show[p4, p1, p2, AspectRatio \[Rule] Automatic, PlotRange \[Rule] {\(-0.1\), 1.1}, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Ticks \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 600, DisplayFunction -> $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(f[{x1_, x2_}] := {x1\ x2\^2, x1 + x2}\), "\[IndentingNewLine]", \(curlf[{x1_, x2_}] := \[PartialD]\_x1\( f[{x1, x2}]\)[\([2]\)] - \[PartialD]\_x2\( f[{x1, x2}]\)[\([1]\)]\), "\[IndentingNewLine]", \(\(\(curlf[{x1, x2}] /. Thread[{x1, x2} \[Rule] {x\_1, x\_2}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \ \(\(\(Integrate[ Boole[x1\^2 \[LessEqual] x2 \[LessEqual] x1] curlf[{x1, x2}], {x2, 0, 1}, {x1, 0, 1}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(Integrate[f[\[Phi]2[t]] . \(\[Phi]2'\)[t], {t, 0, 1}] + Integrate[f[\[Phi]1[t]] . \(\[Phi]1'\)[t], {t, 1, 0}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 8.5 (Quadrature of parabola) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[TextData[StyleBox["See also the file Ex8.5.pdf.", FontVariations->{"CompatibilityType"->0}]], "Text", Background->RGBColor[0.988235, 0.996078, 0.839216]], Cell[BoxData[ \(\(\( (*\ Mathematica\ does\ not\ treat\ the\ symbols\ \(t\_-\), \ t\_0\ and\ \(t\_+\)\ in\ the\ way\ one\ would\ hope . \ Therefore\ they\ are\ replaced\ by\ tm, \ tz\ and\ tp\ and\ substituted\ back\ if\ necessary\ *) \)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\(Clear["\<`*\>"]\ \[IndentingNewLine] \(lightblue = RGBColor[220/256, 248/256, 248/256];\)\[IndentingNewLine] \(darkblue = RGBColor[34/256, 44/256, 162/256];\)\[IndentingNewLine] \(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\[IndentingNewLine] Print["\"]\[IndentingNewLine] \[Phi][t_] := {t, t\^2}\[IndentingNewLine] \(tm = \(-\(3\/4\)\);\)\[IndentingNewLine] \(tz = \(-\(1\/4\)\);\)\[IndentingNewLine] \(tp = 4\/3;\)\[IndentingNewLine] \(pl = ParametricPlot[\[Phi][t], {t, \(-1\), 3/2}, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] False, AxesStyle \[Rule] RGBColor[0, 0.5, 0], Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine] \(pg = Graphics[{lightgreen, Polygon[{\[Phi][tm], \[Phi][tp], \[Phi][tz], \[Phi][ tm]}]}];\)\[IndentingNewLine] \(li = Graphics[{darkblue, Line[{\[Phi][tm], \[Phi][tp], \[Phi][tz], \[Phi][ tm]}]}];\)\[IndentingNewLine] \(po = Graphics[{darkblue, Point[\[Phi][tm]], Point[\[Phi][tp]], Point[\[Phi][tz]], Point[\[Phi][tm]]}];\)\[IndentingNewLine] \(Show[pl, pg, li, po, DisplayFunction \[Rule] $DisplayFunction];\)\[IndentingNewLine] Clear[tm, tp, tz]\[IndentingNewLine]\[IndentingNewLine] mat[t_] := Transpose[{\[Phi][tp] - \[Phi][tm], \[Phi][tp] - \[Phi][ t]}]\[IndentingNewLine] mat[t] /. {tp \[Rule] \(t\_+\), tm \[Rule] \(t\_-\)}\[IndentingNewLine] Print["\"]\[IndentingNewLine] hd[t_] := 1\/2\ Det[mat[t]]\[IndentingNewLine] hd[t] /. {tp \[Rule] \(t\_+\), tm \[Rule] \(t\_-\)}\[IndentingNewLine] Factor[hd[t]] /. {tp \[Rule] \(t\_+\), tm \[Rule] \(t\_-\)}\[IndentingNewLine] sol = \(\(Solve[D[hd[t], t] \[Equal] 0, t]\)[\([1, 1]\)] /. {Rule \[Rule] Equal}\) /. {tp \[Rule] \(t\_+\), tm \[Rule] \(t\_-\)}\[IndentingNewLine] Simplify[hd[t], sol] /. {tp \[Rule] \(t\_+\), tm \[Rule] \(t\_-\)}\[IndentingNewLine] \(\[Delta] = \(\(t\_+\) - \(t\_-\)\)\/2;\)\[IndentingNewLine] Simplify[\((Simplify[hd[t], sol] /. {tp \[Rule] \(t\_+\), tm \[Rule] \(t\_-\)})\) \[Equal] \[Delta]^3]\[IndentingNewLine]\ \[IndentingNewLine] Print["\"]\[IndentingNewLine] \(\(\[Phi]'\)[t]\)[\([2]\)]\/\(\(\[Phi]'\)[t]\)[\([1]\)]\ \[IndentingNewLine] Cancel[\((\[Phi][\(t\_+\)] - \ \[Phi][\(t\_-\)])\)[\([2]\)]\/\((\[Phi][\(t\_+\)] - \[Phi][\(t\_-\)])\)[\([1]\ \)]]\[IndentingNewLine] %% == %\[IndentingNewLine] \(pl = ParametricPlot[\[Phi][t], {t, \(-1\), 3\/2}, PlotStyle \[Rule] RGBColor[1, 0, 0], Axes \[Rule] False, Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine] \(tm = \(-\(3\/4\)\);\)\[IndentingNewLine] \(tp = 4\/3;\)\[IndentingNewLine] \(tz = \(1\/2\) \((tp + tm)\);\)\[IndentingNewLine] \(li = Graphics[{darkblue, Line[{\[Phi][tm], \[Phi][tp], \[Phi][tz], \[Phi][ tm]}]}];\)\[IndentingNewLine] \(ta = Graphics[{darkblue, Line[{\[Phi][tz] - \(1\/2\) \(\[Phi]'\)[tz], \[Phi][ tz] + \(\[Phi]'\)[tz]}]}];\)\[IndentingNewLine] \(po = Graphics[{darkblue, Point[\[Phi][tm]], Point[\[Phi][tp]], Point[\[Phi][tz]], Point[\[Phi][tm]]}];\)\[IndentingNewLine] \(Show[pl, li, ta, po, DisplayFunction \[Rule] $DisplayFunction];\)\[IndentingNewLine]\ \[IndentingNewLine] Print["\"]\[IndentingNewLine] \(plo = ParametricPlot[\[Phi][t], {t, tm, tp}, PlotStyle \[Rule] lightgreen, DisplayFunction \[Rule] Identity];\)\[IndentingNewLine] \(se = plo /. \((Line[pts_] \[RuleDelayed] Polygon[pts])\);\)\[IndentingNewLine] \(Show[se, pl, li, po, PlotRange \[Rule] All, Axes \[Rule] False, Background \[Rule] lightblue, Ticks \[Rule] None, ImageSize \[Rule] 600, DisplayFunction \[Rule] $DisplayFunction];\)\[IndentingNewLine] Clear[tm, tp, tz]\[IndentingNewLine] \(t\_0 = \(\(t\_+\) + \(t\_-\)\)\/2;\)\[IndentingNewLine] Simplify[\(1\/3\) \((\(t\_+\)\^3 - \(t\_-\)\^3)\) \[Equal] \(2\/3\) \ \[Delta]\^3 + 2 \( t\_0\^2\) \[Delta]]\[IndentingNewLine] Print["\"]\[IndentingNewLine] Integrate[ 1, {x1, \(t\_-\), \(t\_+\)}, {x2, x1\^2, \((t\_0 - \[Delta])\)\^2 + 2 \( t\_0\) \((x1 - \(t\_-\))\)}]\[IndentingNewLine] Simplify[% \[Equal] \(4\/3\) \[Delta]\^3]\[IndentingNewLine] Print["\"]\[IndentingNewLine] i[t_] := \(\[Phi][t]\)[\([1]\)] \(\(\[Phi]'\)[t]\)[\([2]\)] - \(\[Phi][ t]\)[\([2]\)] \(\(\[Phi]'\)[t]\)[\([1]\)]\[IndentingNewLine] y[t_] := \[Phi][tp] + t \((\[Phi][tm] - \[Phi][tp])\)\[IndentingNewLine] ii = Simplify[\(y[t]\)[\([1]\)] \(\(y'\)[t]\)[\([2]\)] - \(y[ t]\)[\([2]\)] \(\(y'\)[ t]\)[\([1]\)]] /. {tp \[Rule] \(t\_+\), tm \[Rule] \(t\_-\)}\[IndentingNewLine] Simplify[\(1\/2\) \(\[Integral]\_\(t\_-\)\%\(t\_+\)i[ t] \[DifferentialD]t\) + \(1\/2\) ii]\[IndentingNewLine] Print["\"]\[IndentingNewLine] Simplify[\(1\/2\) \((\(\[Phi][\(t\_-\)]\)[\([2]\)] + \ \(\[Phi][\(t\_+\)]\)[\([2]\)])\) \((\(t\_+\) - \(t\_-\))\) - \[Integral]\_\(t\ \_-\)\%\(t\_+\)\(t\^2\) \[DifferentialD]t]\[IndentingNewLine]\ \[IndentingNewLine] Print["\"]\ \[IndentingNewLine] \[CapitalPsi][{y1_, y2_}] := {t\_0 + \[Delta]\ y1, t\_0\^2 + 2 t\_0\ \[Delta]\ y1 + \[Delta]\^2\ y2}\[IndentingNewLine] Simplify[\[CapitalPsi][\[Phi][t]] == \[Phi][ t\_0 + \[Delta]\ t]]\[IndentingNewLine] Simplify[ Map[\[CapitalPsi], {{\(-1\), 1}, {0, 0}, {1, 1}}] == {\[Phi][\(t\_-\)], \[Phi][ t\_0], \[Phi][\(t\_+\)]}]\[IndentingNewLine]\[IndentingNewLine] Print["\"]\ \[IndentingNewLine] Print["\"]\[IndentingNewLine] \((1 - \((\(-1\))\))\) - \[Integral]\_\(-1\)\%1\( t\^2\) \[DifferentialD]t\)\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 8.7 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(<< Calculus`VectorAnalysis`\), "\n", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(darkblue = RGBColor[34/256, 44/256, 162/256];\)\), "\[IndentingNewLine]", \(\(lightgreen = RGBColor[220/256, 248/256, 230/256];\)\), "\n", \(\[Psi][\[Theta]_, z_] := {Cos[\[Theta]], Sin[\[Theta]], z}\), "\[IndentingNewLine]", \(\[Phi][x_, y_] := {x, y, 1 - x - y}\), "\[IndentingNewLine]", \(\(cylinderplot = ParametricPlot3D[ Append[\[Psi][\[Theta], z], SurfaceColor[ RGBColor[0.7, 0.7, 1]]], {\[Theta], \(-\[Pi]\), \[Pi]}, {z, \(-1\), 2.5}, DisplayFunction \[Rule] Identity, ViewPoint \[Rule] {2, 1, 0.9}, PlotPoints \[Rule] 50];\)\), "\n", \(\(planeplot = ParametricPlot3D[ Append[\[Phi][x, y], SurfaceColor[RGBColor[0.7, 0.7, 1]]], {x, \(-1\), 1}, {y, \(-1\), 1}, DisplayFunction \[Rule] Identity, PlotPoints \[Rule] 50];\)\), "\[IndentingNewLine]", \(\(Show[cylinderplot, planeplot, ViewPoint \[Rule] {2, 0.5, 1.9}, DisplayFunction \[Rule] $DisplayFunction, Ticks \[Rule] False, Boxed \[Rule] False, Axes \[Rule] None, Background \[Rule] lightblue, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, ImageSize \[Rule] 500, AspectRatio \[Rule] 1.2];\)\), "\[IndentingNewLine]", \(f[x_] := \ {\(-x\_\(\(\[LeftDoubleBracket]\)\(2\)\(\[RightDoubleBracket]\)\)\^3\), x\_\(\(\[LeftDoubleBracket]\)\(1\)\(\[RightDoubleBracket]\)\)\^3, x\_\(\(\[LeftDoubleBracket]\)\(3\)\(\[RightDoubleBracket]\)\)\^3}\), \ "\n", \(Print["\"]\), "\ \[IndentingNewLine]", \(curlf[{x_, y_, z_}] = Curl[f[{x, y, z}], Cartesian[x, y, z]]\), "\[IndentingNewLine]", \(\[Phi]normal[x_, y_] := \(-D[\[Phi][x, y], x]\)\[Times] D[\[Phi][x, y], y]\), "\[IndentingNewLine]", \(curlf[\[Phi][x, y]] . \[Phi]normal[x, y]\), "\[IndentingNewLine]", \(\[Integral]\_\(-1\)\%1\(\[Integral]\_\(-\@\(1 - x\^2\)\)\%\(\@\(1 - \ x\^2\)\)curlf[\[Phi][x, y]] . \[Phi]normal[x, y] \[DifferentialD]y \[DifferentialD]x\)\), \ "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Gamma][a_] := {Cos[a], \(-Sin[a]\), 1 - Cos[a] + Sin[a]}\), "\n", \(\[Gamma][0]\), "\[IndentingNewLine]", \(\(\(\[Gamma]'\)[0]\)\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)\), "\n", \(f[\[Gamma][\[Alpha]]] . \(\[Gamma]'\)[\[Alpha]]\), \ "\[IndentingNewLine]", \(\[Integral]\_0\%\(2 \[Pi]\)f[\[Gamma][\[Alpha]]] . \(\[Gamma]'\)[\ \[Alpha]] \[DifferentialD]\[Alpha]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 8.10 ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(<< Calculus`VectorAnalysis`\), "\[IndentingNewLine]", \(\(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\[Phi][\[Alpha]_, \[Theta]_] := {Cos[\[Alpha]] Cos[\[Theta]], Sin[\[Alpha]] Cos[\[Theta]], Sin[\[Theta]]}\), "\[IndentingNewLine]", \(\(\[Psi] = \[Pi]\/8;\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[ Append[\[Phi][\[Alpha], \[Theta]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \[Psi], \ \[Pi]\/2}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, ViewPoint \[Rule] {2, 1, 0.9}, PlotPoints \[Rule] {71, 31}, Ticks \[Rule] False, Boxed \[Rule] False, Axes \[Rule] None, Background \[Rule] lightblue, ImageSize \[Rule] 600];\)\), "\[IndentingNewLine]", \(\(\(Clear[\[Psi]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(g[{x1_, x2_, x3_}] := \(x3\/\(x1\^2 + x2\^2\)\) {x2, \(-x1\), 0}\), "\[IndentingNewLine]", \(\[Gamma][\[Alpha]_] := {Cos[\[Alpha]] Cos[\[Psi]], Sin[\[Alpha]] Cos[\[Psi]], Sin[\[Psi]]}\), "\[IndentingNewLine]", \(\(\(Integrate[ Simplify[\(Composition[ g, \[Gamma]]\)[\[Alpha]] . \(\[Gamma]'\)[\[Alpha]]], {\[Alpha], \ \(-\[Pi]\), \[Pi]}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(rotg[x\_1, x\_2, x\_3] = Simplify[Curl[g[{x1, x2, x3}], Cartesian[x1, x2, x3]]] /. Thread[{x1, x2, x3} -> {x\_1, x\_2, x\_3}]\), "\[IndentingNewLine]", \(\(\(Simplify[ rotg[x\_1, x\_2, x\_3] . {x\_1, x\_2, x\_3}]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]]\), "\ \[IndentingNewLine]", \(\[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]]\), "\ \[IndentingNewLine]", \(c = Simplify[\[PartialD]\_\[Alpha]\ \[Phi][\[Alpha], \[Theta]]\[Cross]\ \[PartialD]\_\[Theta]\ \[Phi][\[Alpha], \[Theta]]]\), "\[IndentingNewLine]", \(Simplify[\@\(\[Sum]\+\(j = 1\)\%3 c[\([j]\)]\^2\), Assumptions -> {Im[\[Alpha]] \[Equal] 0, \(-\(\[Pi]\/2\)\) < \[Theta] < \[Pi]\/2}]\), "\ \[IndentingNewLine]", \(Integrate[%, {\[Alpha], \(-\[Pi]\), \[Pi]}, {\[Theta], \[Psi], \ \[Pi]\/2}]\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["NEW EXERCISES", FontWeight->"Bold"]], "Subtitle", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[CellGroupData[{ Cell["New Exercise 7.0 (Lambert's projection) ", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\(step = 0.8;\)\), "\n", \(\(viewpoint = {8, 10, 4};\)\), "\n", \(\(c[x_, \[Alpha]_] := {x, Cos[\[Alpha]], Sin[\[Alpha]]};\)\), "\n", \(\(cyl = ParametricPlot3D[ Evaluate[ Append[c[x, \[Alpha]], SurfaceColor[RGBColor[0.7, 0.7, 1]]]], {x, \(-1\), 1}, {\[Alpha], \(-\[Pi]\), 0}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{3, 5, 2}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] 68, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(s[x_] := \@\(1 - x\^2\)\), "\[IndentingNewLine]", \(\(\[Phi][x_, \[Alpha]_] := {x, s[x]\ Cos[\[Alpha]], s[x]\ Sin[\[Alpha]]};\)\), "\[IndentingNewLine]", \(\(sph = ParametricPlot3D[ Evaluate[ Append[\[Phi][x, \[Alpha]], SurfaceColor[RGBColor[0.7, 0.7, 0]]]], {x, \(-1\), 1}, {\[Alpha], \(-\[Pi]\), 0}, LightSources \[Rule] {{1/4 viewpoint, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, PlotPoints \[Rule] 65, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(lines = Graphics3D[ Table[{RGBColor[1, 0, 0], Line[{{x, 0, 0}, 1.1\ c[x, \[Alpha]]}]}, {x, \(-1.6\) + step, 1.6 - step, step}, {\[Alpha], \(-\[Pi]\), 0, \[Pi]/30}]];\)\), "\[IndentingNewLine]", \(\(ax = Graphics3D[{RGBColor[1, 0, 0], Thickness[0.004], Line[{1.2 {\(-1\), 0, 0}, 1.2 {1, 0, 0}}]}];\)\), "\[IndentingNewLine]", \(\(\(Show[cyl, sph, lines, ax, ViewPoint \[Rule] viewpoint, AspectRatio \[Rule] Automatic, Background \[Rule] lightblue, Axes \[Rule] False, Boxed \[Rule] False, Ticks \[Rule] None, ImageSize \[Rule] 1200, DisplayFunction \[Rule] $DisplayFunction];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(\(<< Miscellaneous`WorldPlot`;\)\), "\n", \(\(worldmap = WorldPlot[{World, RandomColors}, WorldProjection \[Rule] LambertCylindrical, WorldPoints \[Rule] 2000, WorldRange \[Rule] {{\(-60\), 90}, {\(-180\), 105}}, WorldBackground \[Rule] GrayLevel[0.96], WorldGrid \[Rule] {7.5, 7.5}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[{worldmap}, ImageSize \[Rule] 1200, Background \[Rule] lightblue, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ New Exercise 7.1 (Frenet-Serret formulae and tubular \ neighborhoods) \ \>", "Section", FontColor->RGBColor[0.0156252, 0.191409, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[BoxData[{\(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Off[ General::spell1]\), "\[IndentingNewLine]", \(lightblue = RGBColor[220/256, 248/256, 248/256];\), "\[IndentingNewLine]", \(darkblue = RGBColor[34/256, 44/256, 162/256];\), "\[IndentingNewLine]", \(Print[ Helix]\), "\[IndentingNewLine]", \(\[Gamma][ s_] := \(1\/\@2\) {\(-s\), Cos[s], Sin[s]}\), "\[IndentingNewLine]", \(t[s_] := \(\[Gamma]'\)[ s]\), "\[IndentingNewLine]", \(\(\[Gamma]''\)[ s]\), "\[IndentingNewLine]", \(n[ s_] := \(\@2\) \(\[Gamma]''\)[ s]\), "\[IndentingNewLine]", \(Simplify[\@\(\[Sum]\+\(j = \ 1\)\%3\(\( \[Gamma]'\)[s]\)[\([j]\)]\^2\)]\), "\[IndentingNewLine]", \ \(Simplify[\@\(\[Sum]\+\(j = 1\)\%3\( n[s]\)[\([j]\)]\^2\)]\), "\ \[IndentingNewLine]", \(b[s_] := Cross[t[s], n[s]]\), "\[IndentingNewLine]", \(view = 3 {1, \(-2\), 3};\), "\[IndentingNewLine]", \(val = 0;\), "\[IndentingNewLine]", \(plane = ParametricPlot3D[ Append[\[Gamma][val] + \@r\ Cos[\[Alpha]] n[val] + \(\@r\) Sin[\[Alpha]] b[val], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {r, 0, 0.6}, {\[Alpha], \(-\[Pi]\), \[Pi]}, PlotPoints \[Rule] {12, 31}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Axes \[Rule] None, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, Ticks \[Rule] None, ViewPoint \[Rule] view, Background \[Rule] lightblue, ImageSize \[Rule] 800, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(h = ParametricPlot3D[ Append[\[Gamma][s], RGBColor[1, 0, 0] // Evaluate], {s, \(-\(\[Pi]\/2\)\), \[Pi]\/2}, Axes \[Rule] None, PlotRange \[Rule] All, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, Ticks \[Rule] None, PlotPoints \[Rule] 200, ViewPoint \[Rule] view, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(tan = Graphics3D[{RGBColor[0, 0.5, 0], Line[{\[Gamma][s], \[Gamma][s] + t[s]}] /. s \[Rule] val}];\), "\[IndentingNewLine]", \(nor = Graphics3D[{RGBColor[0, 0.5, 0], Line[{\[Gamma][s], \[Gamma][s] + n[s]}] /. s \[Rule] val}];\), "\[IndentingNewLine]", \(bin = Graphics3D[{RGBColor[0, 0.5, 0], Line[{\[Gamma][s], \[Gamma][s] + b[s]}] /. s \[Rule] val}];\), "\[IndentingNewLine]", \(p1 = Graphics3D[{darkblue, Point[\[Gamma][s] + t[s] /. s \[Rule] val]}];\), "\[IndentingNewLine]", RowBox[{\(p2 = Graphics3D[{darkblue, Point[\[Gamma][s] + n[s] /. s \[Rule] val]}];\), " "}], "\[IndentingNewLine]", \(p3 = Graphics3D[{darkblue, Point[\[Gamma][s] + b[s] /. s \[Rule] val]}];\), "\[IndentingNewLine]", RowBox[{ RowBox[{"t1", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\[Gamma](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))+\!\(\* StyleBox[\"T\",\nFontSlant->\"Italic\"]\)(\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", ",", \(\[Gamma][s] + t[s] + {\(-0.05\), 0.05, 0.15} /. s \[Rule] val\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"t2", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\[Gamma](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))+\!\(\* StyleBox[\"N\",\nFontSlant->\"Italic\"]\)(\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", ",", \(\[Gamma][s] + n[s] + {\(-0.05\), 0.05, \(-0.3\)} /. s \[Rule] val\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"t3", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\[Gamma](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))+\!\(\* StyleBox[\"B\",\nFontSlant->\"Italic\"]\)(\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", ",", \(\[Gamma][s] + b[s] + {0.3, 0.05, 0.12} /. s \[Rule] val\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", \(t4 = Graphics3D[ Text["\", \[Gamma][\(-\[Pi]\)/3] + {0.12, 0.1, 0.1} /. s \[Rule] val]];\), "\[IndentingNewLine]", RowBox[{ RowBox[{"t5", "=", RowBox[{"Graphics3D", "[", RowBox[{"Text", "[", RowBox[{"\"\<\[UpTee](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", ",", \(\((\[Gamma][s] + 0.7 n[s] + 0.7 b[s])\) + {0.3, 0.1, 1.8} /. s \[Rule] val\)}], "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", \(Show[plane, tan, nor, bin, p1, p2, p3, t1, t2, t3, t4, t5, h, TextStyle \[Rule] {FontFamily -> "\", FontSize \[Rule] 14. }, DisplayFunction \[Rule] $DisplayFunction];\), "\[IndentingNewLine]", \ \(r = 0.3;\), "\[IndentingNewLine]", \(view = {\(-1\), \(-2\), 0};\), "\[IndentingNewLine]", \(tube = ParametricPlot3D[ Append[\[Gamma][s] + r\ Cos[\[Alpha]] n[s] + r\ Sin[\[Alpha]] b[s], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, \(-\(\[Pi]\/2\)\) - 0.5, \[Pi]\/2 + 0.5}, {\[Alpha], \(-\[Pi]\), \[Pi]}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {80, 60}, Background \[Rule] lightblue, ImageSize \[Rule] 800];\), "\[IndentingNewLine]", \(halftube = ParametricPlot3D[ Append[\[Gamma][s] + r\ Cos[\[Alpha]] n[s] + r\ Sin[\[Alpha]] b[s], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, \(-\(\[Pi]\/2\)\) - 0.5, \[Pi]\/2 + 0.5}, {\[Alpha], \[Pi]/2, 3 \[Pi]/2}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {80, 30}, Background \[Rule] lightblue, ImageSize \[Rule] 800, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(cur = ParametricPlot3D[ Append[\[Gamma][s], RGBColor[1, 0, 0] // Evaluate], {s, \(-\(\[Pi]\/2\)\) - 0.5, \[Pi]\/2 + 0.5}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] 30, ImageSize \[Rule] 800, DisplayFunction \[Rule] Identity];\), "\[IndentingNewLine]", \(curve = cur /. \((Line[pts_] \[RuleDelayed] {Thickness[0.004], Line[pts]})\);\), "\[IndentingNewLine]", RowBox[{\(Show[halftube, curve, ImageSize \[Rule] 800, DisplayFunction \[Rule] $DisplayFunction];\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", \(Clear["\<`*\>"]\), "\ \[IndentingNewLine]", \(Print[ Helix]\), "\[IndentingNewLine]", \(\[Gamma][ s_] := \(1\/\@2\) {Cos[s], Sin[s], s}\), "\[IndentingNewLine]", \(t[ s_] := \(\[Gamma]'\)[s]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\!\(\* StyleBox[\"T\",\nFontSlant->\"Italic\"]\)(\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(\(1\/\@2\) {\(-Sin[s]\), Cos[s], 1}\), "\[IndentingNewLine]", \(n[ s_] := \(\@2\) \(\[Gamma]''\)[s]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\!\(\* StyleBox[\"N\",\nFontSlant->\"Italic\"]\)(\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(n[ s]\), "\[IndentingNewLine]", \(Simplify[\@\(\[Sum]\+\(j = 1\)\%3\(\( \ \[Gamma]'\)[s]\)[\([j]\)]\^2\)]\), "\[IndentingNewLine]", \(Simplify[\@\(\ \[Sum]\+\(j = 1\)\%3\( n[s]\)[\([j]\)]\^2\)]\), "\[IndentingNewLine]", \(b[ s_] := Cross[t[s], n[s]]\), "\[IndentingNewLine]", \(Simplify[ b[s] == \(1\/\@2\) {Sin[s], \(-Cos[s]\), 1}]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\!\(\* StyleBox[\"B\",\nFontSlant->\"Italic\"]\)(\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(b[ s_] := \(1\/\@2\) {Sin[s], \(-Cos[s]\), 1}\), "\[IndentingNewLine]", \(b[s]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\[Kappa](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(Simplify[\(t'\)[s] . n[s]]\), "\[IndentingNewLine]", RowBox[{"Print", "[", "\"\<\[Tau](\!\(\* StyleBox[\"s\",\nFontSlant->\"Italic\"]\))\>\"", "]"}], "\[IndentingNewLine]", \(Simplify[\(-\(b'\)[s] . n[s]\)]\)}], "Input"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Gamma][ s_] := \(1\/\@2\) {Cos[s], Sin[s], s}\), "\[IndentingNewLine]", \(t[s_] := \(\[Gamma]'\)[s]\), "\[IndentingNewLine]", \(n[s_] := \(\@2\) \(\[Gamma]''\)[s]\), "\[IndentingNewLine]", \(b[s_] := \(1\/\@2\) {Sin[s], \(-Cos[s]\), 1}\), "\[IndentingNewLine]", \(f1[s_, t_] := \[Gamma][s] + t\ n[s]\), "\[IndentingNewLine]", \(f2[s_, t_] := \[Gamma][s] + t\ n[s] + b[s]\), "\[IndentingNewLine]", \(f3[s_, u_] := \[Gamma][s] + u\ b[s]\), "\[IndentingNewLine]", \(f4[s_, u_] := \[Gamma][s] + n[s] + u\ b[s]\), "\[IndentingNewLine]", \(f5[s_, u_] := \[Gamma][\[Pi]] + t\ n[\[Pi]] + u\ b[\[Pi]]\), "\[IndentingNewLine]", \(\(t1 = ParametricPlot3D[ Append[f1[s, t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, \(-\[Pi]\), \[Pi]}, {t, 0, 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {120, 40}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(t2 = ParametricPlot3D[ Append[f2[s, t], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, \(-\[Pi]\), \[Pi]}, {t, 0, 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {120, 40}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(t3 = ParametricPlot3D[ Append[f3[s, u], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, \(-\[Pi]\), \[Pi]}, {u, 0, 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {120, 40}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(t4 = ParametricPlot3D[ Append[f4[s, u], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, \(-\[Pi]\), \[Pi]}, {u, 0, 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {120, 40}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(t5 = ParametricPlot3D[ Append[f5[t, u], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {t, 0, 1}, {u, 0, 1}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {40, 40}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(c0 = ParametricPlot3D[ Append[\[Gamma][s], RGBColor[1, 0, 0] // Evaluate], {s, \(-\[Pi]\), \[Pi]}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] 120, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(c1 = c0 /. \((Line[pts_] \[RuleDelayed] {Thickness[0.006], Line[pts]})\);\)\), "\[IndentingNewLine]", \(\(view = {1, \(-2\), 0};\)\), "\[IndentingNewLine]", \(\(Show[t1, t2, t3, t4, t5, c1, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\[Gamma][ s_] := \(1\/\@2\) {Cos[s], Sin[s], s/2}\), "\[IndentingNewLine]", \(t[s_] := \(\[Gamma]'\)[s]\), "\[IndentingNewLine]", \(n[s_] := \(\@2\) \(\[Gamma]''\)[s]\), "\[IndentingNewLine]", \(b[s_] := Cross[t[s], n[s]]\), "\[IndentingNewLine]", \(f1[s_, \[Alpha]_] := \[Gamma][s] + 0.2 Cos[ s] \((\((Cos[\[Alpha]] - 1\ )\) n[s] + Sin[\[Alpha]] b[s])\)\), "\[IndentingNewLine]", \(\(t1 = ParametricPlot3D[ Append[f1[s, \[Alpha]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, \(-2\) \[Pi], 2 \[Pi]}, {\[Alpha], \(-\[Pi]\), \[Pi]}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {160, 30}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(c0 = ParametricPlot3D[ Append[\[Gamma][s], RGBColor[1, 0, 0] // Evaluate], {s, \(-\[Pi]\), \[Pi]}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] 120, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(c1 = c0 /. \((Line[pts_] \[RuleDelayed] {Thickness[0.006], Line[pts]})\);\)\), "\[IndentingNewLine]", \(\(view = {1, \(-2\), 0};\)\), "\[IndentingNewLine]", \(\(Show[t1, c1, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(\[Gamma][s_] := {\((4 - s)\) Cos[\[Pi]\ s], \((4 - s)\) Sin[\[Pi]\ s], s}\), "\[IndentingNewLine]", \(t[s_] := FullSimplify[\(1\/Norm[\(\[Gamma]'\)[s]]\) \(\[Gamma]'\)[s], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(b[s_] := FullSimplify[\(1\/Norm[Cross[\(\[Gamma]'\)[s], \(\[Gamma]''\)[s]]]\) Cross[\(\[Gamma]'\)[s], \(\[Gamma]''\)[s]], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[Norm[b[s]]^2, Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[t[s] . b[s], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(n[s_] := Cross[b[s], t[s]]\), "\[IndentingNewLine]", \(FullSimplify[Norm[n[s]]^2, Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[n[s] . t[s], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[n[s] . b[s], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(\(view = {2, \(-2\), 1};\)\), "\[IndentingNewLine]", \(f[s_, \[Alpha]_] := \[Gamma][ s] + \(1\/2\) \(\@\(4 - s\)\) \((Sin[\[Alpha]] n[s] + Cos[\[Alpha]] b[s])\)\), "\[IndentingNewLine]", \(\(t = ParametricPlot3D[ Append[f[s, \[Alpha]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, 0, 4}, {\[Alpha], \(-\[Pi]\), \[Pi]}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {160, 30}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(c0 = ParametricPlot3D[ Append[\[Gamma][s], RGBColor[1, 0, 0] // Evaluate], {s, 0, 4}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] 120, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(c1 = c0 /. \((Line[pts_] \[RuleDelayed] {Thickness[0.006], Line[pts]})\);\)\), "\[IndentingNewLine]", \(\(Show[t, c1, ImageSize \[Rule] 800, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] $DisplayFunction];\)\), \ "\[IndentingNewLine]", \(Print["\"]\), "\[IndentingNewLine]", \(FullSimplify[ Integrate[\[Pi] \(\( 4 - s\)\/4\) Norm[\(\[Gamma]'\)[s]], {s, 0, 4}]]\)}], "Input"], Cell[BoxData[{ \(Clear["\<`*\>"]\), "\[IndentingNewLine]", \(\(lightblue = RGBColor[220/256, 248/256, 248/256];\)\), "\[IndentingNewLine]", \(Print["\"]\), "\ \[IndentingNewLine]", \(\[Gamma][s_] := {Cos[s], 2 Sin[s], 0}\), "\[IndentingNewLine]", \(t[s_] := FullSimplify[\(1\/Norm[\(\[Gamma]'\)[s]]\) \(\[Gamma]'\)[s], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(b[s_] := FullSimplify[\(1\/Norm[Cross[\(\[Gamma]'\)[s], \(\[Gamma]''\)[s]]]\) Cross[\(\[Gamma]'\)[s], \(\[Gamma]''\)[s]], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[Norm[b[s]]^2, Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[t[s] . b[s], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(n[s_] := Cross[b[s], t[s]]\), "\[IndentingNewLine]", \(FullSimplify[Norm[n[s]]^2, Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[n[s] . t[s], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(FullSimplify[n[s] . b[s], Im[s] \[Equal] 0]\), "\[IndentingNewLine]", \(\(r = 0.7;\)\), "\[IndentingNewLine]", \(\(view = {0, \(-3\), 1.5};\)\), "\[IndentingNewLine]", \(f[s_, \[Alpha]_] := \[Gamma][s] + r \((Cos[\[Alpha]] n[s] + Sin[\[Alpha]] b[s])\)\), "\[IndentingNewLine]", \(\(t = ParametricPlot3D[ Append[f[s, \[Alpha]], SurfaceColor[RGBColor[0.7, 0.7, 1]] // Evaluate], {s, \(-\[Pi]\), \[Pi]}, {\[Alpha], \(-\[Pi]\), \ \[Pi]}, LightSources \[Rule] {{{1, 0, 0}, RGBColor[1, 0, 0]}, {{1, 0, 0}, RGBColor[1, 0, 0]}, {{\(-1\), 0, 0}, RGBColor[1, 0, 0]}, {{0, 1, 0}, RGBColor[1, 1, 0]}, {{0, 2, 2}, RGBColor[0, 1, 1]}}, AspectRatio \[Rule] Automatic, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] {160, 30}, Background \[Rule] lightblue, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(c0 = ParametricPlot3D[ Append[\[Gamma][s], RGBColor[1, 0, 0] // Evaluate], {s, \(-\[Pi]\), \[Pi]}, Boxed \[Rule] False, Axes \[Rule] None, ViewPoint \[Rule] view, PlotPoints \[Rule] 120, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(c1 = c0 /. \((Line[pts_] \[RuleDelayed] {Thickness[0.006], Line[pts]})\);\)\), "\[IndentingNewLine]", \(\(Show[t, c1, ImageSize \[Rule] 500, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"], Cell[BoxData[{ \(TimeUsed[]/60\), "\[IndentingNewLine]", \(MaxMemoryUsed[]\)}], "Input"] }, Open ]] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1600}, {0, 1117}}, AutoGeneratedPackage->Automatic, ScreenStyleEnvironment->"Working", WindowSize->{1400, 929}, WindowMargins->{{43, Automatic}, {11, Automatic}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, ShowSelection->True, AnimationDisplayTime->0.2, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. Make modifications to any definition using \ commands in the Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, CellLabelMargins->{{12, Inherited}, {Inherited, Inherited}}, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, CellLabelMargins->{{24, Inherited}, {Inherited, Inherited}}, ScriptMinSize->12], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}}, ScriptMinSize->8], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, CellLabelMargins->{{2, Inherited}, {Inherited, Inherited}}, ScriptMinSize->5, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellMargins->{{12, Inherited}, {20, 40}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{1, 11}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subtitle", 0}, {"Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->36, FontWeight->"Bold"], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, 10}, {20, 40}}, LineSpacing->{1, 0}, FontSize->44], Cell[StyleData["Title", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, FontSize->20], Cell[StyleData["Title", "Printout"], CellMargins->{{2, 10}, {12, 30}}, FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{12, Inherited}, {20, 15}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->24], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontSize->36], Cell[StyleData["Subtitle", "Condensed"], CellMargins->{{8, 10}, {4, 4}}, FontSize->14], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellMargins->{{12, Inherited}, {20, 15}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Helvetica", FontSize->14, FontSlant->"Italic"], Cell[StyleData["Subsubtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Subsubtitle", "Condensed"], CellMargins->{{8, 10}, {8, 8}}, FontSize->12], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[FilledSquare]", CellMargins->{{25, Inherited}, {8, 24}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{1, 7}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->16, FontWeight->"Bold"], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{13, 0}, {7, 22}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 20}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{36, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->22], Cell[StyleData["Subsection", "Condensed"], CellMargins->{{16, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Subsection", "Printout"], CellMargins->{{9, 0}, {7, 22}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 18}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, DefaultNewInlineCellStyle->"None", InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, LanguageCategory->"NaturalLanguage", CounterIncrements->"Subsubsection", FontFamily->"Times", FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{34, 10}, {11, 26}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{17, Inherited}, {6, 12}}, FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{9, 0}, {7, 14}}, FontSize->11] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[StyleData["Text"], CellMargins->{{12, 10}, {7, 7}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, Hyphenation->True, LineSpacing->{1, 3}, CounterIncrements->"Text", FontFamily->"Helvetica", FontColor->RGBColor[0.0195315, 0.17969, 0.55079], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[RawData["\<\ Cell[StyleData[\"Text\", \"Presentation\"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16]\ \>"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[RawData["\<\ Cell[StyleData[\"Text\", \"Condensed\"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11]\ \>"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[RawData["\<\ Cell[StyleData[\"Text\", \"Printout\"], CellMargins->{{2, 2}, {6, 6}}, TextJustification->0.5, FontSize->10]\ \>"], CellMargins->{{2, 2}, {6, 6}}, TextJustification->0.5, FontSize->10], Cell[RawData["\<\ Cell[StyleData[\"SmallText\"], CellMargins->{{12, 10}, {6, 6}}, DefaultNewInlineCellStyle->\"None\", Hyphenation->True, LineSpacing->{1, 3}, LanguageCategory->\"NaturalLanguage\", CounterIncrements->\"SmallText\", FontFamily->\"Helvetica\", FontSize->9, FontColor->RGBColor[0, 0, 1]]\ \>"], CellMargins->{{12, 10}, {6, 6}}, DefaultNewInlineCellStyle->"None", Hyphenation->True, LineSpacing->{1, 3}, LanguageCategory->"NaturalLanguage", CounterIncrements->"SmallText", FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[RawData["\<\ Cell[StyleData[\"SmallText\", \"Presentation\"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12]\ \>"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12], Cell[RawData["\<\ Cell[StyleData[\"SmallText\", \"Condensed\"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontSize->9]\ \>"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontSize->9], Cell[RawData["\<\ Cell[StyleData[\"SmallText\", \"Printout\"], CellMargins->{{2, 2}, {5, 5}}, TextJustification->0.5, FontSize->7]\ \>"], CellMargins->{{2, 2}, {5, 5}}, TextJustification->0.5, FontSize->7], Cell[RawData["Cell[\"Styles for Input/Output\", \"Section\"]"], "Section"], Cell[RawData["\<\ Cell[\"\\<\\ The cells in this section define styles used for input and output to the \\ kernel. Be careful when modifying, renaming, or removing these styles, \ because the \\ front end associates special meanings with these style names. Some attributes \ for \\ these styles are actually set in FormatType Styles (in the last section of \ this \\ stylesheet). \\ \\>\", \"Text\"]\ \>"], "Text"], Cell[StyleData["Input"], CellMargins->{{45, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Mathematica", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontWeight->"Bold", FontColor->RGBColor[0, 0, 0.500008], Background->RGBColor[0.832044, 0.996109, 0.832044]], Cell[RawData["\<\ Cell[StyleData[\"Input\", \"Presentation\"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontSize->16]\ \>"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontSize->16], Cell[RawData["\<\ Cell[StyleData[\"Input\", \"Condensed\"], CellMargins->{{40, 10}, {2, 3}}, FontSize->11]\ \>"], CellMargins->{{40, 10}, {2, 3}}, FontSize->11], Cell[RawData["\<\ Cell[StyleData[\"Input\", \"Printout\"], CellMargins->{{39, 0}, {4, 6}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9]\ \>"], CellMargins->{{39, 0}, {4, 6}}, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Mathematica", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", StyleMenuListing->None, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{47, 10}, {7, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}, FontSize->11], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->11, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->11], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->7, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontSize->11], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->10], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageMargins->{{38, Inherited}, {Inherited, 0}}, Magnification->0.6], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->9], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["FrameLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["FrameLabel", "Presentation"], FontSize->12], Cell[StyleData["FrameLabel", "Condensed"], FontSize->9], Cell[StyleData["FrameLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a \ cell exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["TB"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["TBI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["MR"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["MO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["MB"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["MBO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["SR"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "EnhancedPrintout"], StyleMenuListing->None, FontFamily->"Futura", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Outline Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Outline1"], CellMargins->{{12, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 50}, ParagraphIndent->-38, CounterIncrements->"Outline1", FontSize->18, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>CapitalRomanNumeral}], Cell[StyleData["Outline1", "Printout"], CounterBoxOptions->{CounterFunction:>CapitalRomanNumeral}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline2"], CellMargins->{{59, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 60}, ParagraphIndent->-27, CounterIncrements->"Outline2", FontSize->15, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}], Cell[StyleData["Outline2", "Printout"], CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline3"], CellMargins->{{108, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 70}, ParagraphIndent->-21, CounterIncrements->"Outline3", FontSize->12, CounterBoxOptions->{CounterFunction:>Identity}], Cell[StyleData["Outline3", "Printout"], CounterBoxOptions->{CounterFunction:>Identity}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline4"], CellMargins->{{158, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 80}, ParagraphIndent->-18, CounterIncrements->"Outline4", FontSize->10, CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "a", "z"], #]&)}], Cell[StyleData["Outline4", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "Condensed"], FontSize->11], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "Condensed"], FontSize->11], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "Condensed"], FontSize->11], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "Condensed"], FontSize->11], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "Condensed"], FontSize->11], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, Placeholder]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["PrimaryPlaceholder", "Presentation"]], Cell[StyleData["PrimaryPlaceholder", "Condensed"]], Cell[StyleData["PrimaryPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, Hyphenation->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], InputAutoReplacements->{}, AllowInlineCells->False, Hyphenation->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], LanguageCategory->"Formula", ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]], Cell[CellGroupData[{ Cell["Automatic Styles", "Section"], Cell["\<\ The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).\ \>", "Text"], Cell[StyleData["UnmatchedBracket"], StyleMenuListing->None, FontColor->RGBColor[0.760006, 0.330007, 0.8]] }, Closed]] }, Open ]] }] ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{ "UCU311"->{ Cell[3897, 112, 213, 6, 49, "Text", CellTags->"UCU311"], Cell[4113, 120, 116, 3, 46, "Input", InitializationCell->True, CellTags->"UCU311"], Cell[4232, 125, 496, 10, 106, "Input", InitializationCell->True, CellTags->"UCU311"]} } *) (*CellTagsIndex CellTagsIndex->{ {"UCU311", 626694, 14245} } *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 449, 11, 377, "Title"], Cell[2228, 66, 805, 20, 264, "Subtitle"], Cell[3036, 88, 693, 16, 163, "Text"], Cell[CellGroupData[{ Cell[3754, 108, 140, 2, 75, "Section"], Cell[3897, 112, 213, 6, 49, "Text", CellTags->"UCU311"], Cell[4113, 120, 116, 3, 46, "Input", InitializationCell->True, CellTags->"UCU311"], Cell[4232, 125, 496, 10, 106, "Input", InitializationCell->True, CellTags->"UCU311"] }, Closed]], Cell[CellGroupData[{ Cell[4765, 140, 135, 2, 55, "Section"], Cell[4903, 144, 4222, 58, 753, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[9162, 207, 186, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[9373, 214, 167, 5, 102, "Section"], Cell[9543, 221, 1915, 34, 420, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[11495, 260, 172, 5, 82, "Section"], Cell[11670, 267, 367, 9, 49, "Text"], Cell[12040, 278, 1526, 29, 363, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[13603, 312, 139, 2, 55, "Section"], Cell[13745, 316, 355, 11, 52, "Text"], Cell[14103, 329, 886, 19, 302, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[15038, 354, 186, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[15249, 361, 166, 5, 102, "Section"], Cell[15418, 368, 608, 11, 165, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[16063, 384, 151, 2, 55, "Section"], Cell[16217, 388, 2161, 59, 49, "Text"], Cell[18381, 449, 850, 23, 68, "Text"], Cell[19234, 474, 459, 9, 86, "Input"], Cell[19696, 485, 1885, 53, 49, "Text"], Cell[21584, 540, 439, 7, 126, "Input"], Cell[22026, 549, 245, 4, 106, "Input"], Cell[22274, 555, 103, 2, 46, "Input"], Cell[22380, 559, 129, 1, 49, "Text"], Cell[22512, 562, 46, 1, 46, "Input"], Cell[22561, 565, 1812, 51, 49, "Text"], Cell[24376, 618, 775, 14, 266, "Input"], Cell[25154, 634, 1895, 53, 49, "Text"], Cell[27052, 689, 888, 17, 288, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[27977, 711, 174, 2, 55, "Section"], Cell[28154, 715, 237, 3, 49, "Text"], Cell[28394, 720, 81, 1, 46, "Input"], Cell[28478, 723, 275, 6, 49, "Text"], Cell[28756, 731, 5987, 118, 1640, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[34780, 854, 166, 5, 82, "Section"], Cell[34949, 861, 368, 8, 123, "Input"], Cell[35320, 871, 411, 5, 68, "Text"], Cell[35734, 878, 2051, 42, 585, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[37822, 925, 164, 5, 82, "Section"], Cell[37989, 932, 954, 20, 146, "Input"], Cell[38946, 954, 567, 13, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[39550, 972, 192, 5, 82, "Section"], Cell[39745, 979, 353, 6, 122, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[40135, 990, 170, 5, 82, "Section"], Cell[40308, 997, 435, 8, 154, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[40780, 1010, 169, 5, 82, "Section"], Cell[40952, 1017, 368, 6, 117, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[41369, 1029, 186, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[41580, 1036, 191, 5, 102, "Section"], Cell[41774, 1043, 291, 4, 49, "Text"], Cell[42068, 1049, 102, 1, 49, "Text"], Cell[42173, 1052, 2225, 38, 326, "Input"], Cell[44401, 1092, 1773, 36, 266, "Input"], Cell[46177, 1130, 327, 8, 49, "Text"], Cell[46507, 1140, 1735, 34, 394, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[48279, 1179, 162, 5, 82, "Section"], Cell[48444, 1186, 112, 1, 49, "Text"], Cell[48559, 1189, 4177, 75, 466, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[52773, 1269, 162, 5, 82, "Section"], Cell[52938, 1276, 112, 1, 49, "Text"], Cell[53053, 1279, 1621, 29, 286, "Input"], Cell[54677, 1310, 382, 8, 165, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[55096, 1323, 143, 2, 55, "Section"], Cell[55242, 1327, 6519, 151, 612, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[61810, 1484, 186, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[62021, 1491, 160, 2, 75, "Section"], Cell[62184, 1495, 6013, 110, 964, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[68234, 1610, 164, 5, 82, "Section"], Cell[68401, 1617, 104, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[68542, 1623, 157, 2, 55, "Section"], Cell[68702, 1627, 4250, 80, 626, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[72989, 1712, 164, 5, 82, "Section"], Cell[73156, 1719, 2358, 44, 389, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[75551, 1768, 162, 5, 82, "Section"], Cell[75716, 1775, 1836, 35, 267, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[77589, 1815, 172, 5, 82, "Section"], Cell[77764, 1822, 84, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[77885, 1828, 158, 2, 55, "Section"], Cell[78046, 1832, 6291, 118, 806, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[84374, 1955, 150, 2, 55, "Section"], Cell[84527, 1959, 1388, 26, 269, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[85952, 1990, 153, 2, 55, "Section"], Cell[86108, 1994, 2446, 46, 410, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[88591, 2045, 160, 5, 82, "Section"], Cell[88754, 2052, 84, 1, 49, "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[88887, 2059, 186, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[89098, 2066, 164, 5, 102, "Section"], Cell[89265, 2073, 84, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[89386, 2079, 161, 5, 82, "Section"], Cell[89550, 2086, 84, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[89671, 2092, 161, 5, 82, "Section"], Cell[89835, 2099, 84, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[89956, 2105, 165, 5, 82, "Section"], Cell[90124, 2112, 3822, 76, 606, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[93983, 2193, 161, 5, 82, "Section"], Cell[94147, 2200, 117, 1, 49, "Text"], Cell[94267, 2203, 2358, 47, 530, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[96662, 2255, 173, 5, 82, "Section"], Cell[96838, 2262, 4030, 77, 622, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[100905, 2344, 183, 5, 82, "Section"], Cell[101091, 2351, 7887, 132, 1057, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[109015, 2488, 191, 5, 82, "Section"], Cell[109209, 2495, 3222, 61, 624, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[112468, 2561, 169, 2, 55, "Section"], Cell[112640, 2565, 1688, 28, 226, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[114365, 2598, 155, 2, 55, "Section"], Cell[114523, 2602, 11632, 221, 1932, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[126204, 2829, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[126418, 2836, 177, 2, 75, "Section"], Cell[126598, 2840, 178, 5, 49, "Text"], Cell[126779, 2847, 1032, 20, 370, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[127848, 2872, 139, 2, 55, "Section"], Cell[127990, 2876, 187, 5, 49, "Text"], Cell[128180, 2883, 396, 7, 171, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[128613, 2895, 140, 2, 55, "Section"], Cell[128756, 2899, 956, 17, 298, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[129749, 2921, 139, 2, 55, "Section"], Cell[129891, 2925, 191, 5, 49, "Text"], Cell[130085, 2932, 2557, 53, 465, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[132679, 2990, 140, 2, 55, "Section"], Cell[132822, 2994, 644, 12, 128, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[133503, 3011, 231, 5, 55, "Section"], Cell[133737, 3018, 1701, 34, 628, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[135475, 3057, 591, 20, 57, "Section"], Cell[136069, 3079, 355, 7, 171, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[136461, 3091, 177, 2, 55, "Section"], Cell[136641, 3095, 1176, 22, 372, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[137854, 3122, 236, 5, 55, "Section"], Cell[138093, 3129, 391, 9, 68, "Text"], Cell[138487, 3140, 3279, 61, 667, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[141803, 3206, 141, 2, 55, "Section"], Cell[141947, 3210, 414, 10, 49, "Text"], Cell[142364, 3222, 1079, 21, 390, "Input"], Cell[143446, 3245, 103, 3, 49, "Text"], Cell[143552, 3250, 321, 7, 87, "Input"], Cell[143876, 3259, 75, 0, 49, "Text"], Cell[143954, 3261, 1561, 29, 289, "Input"], Cell[145518, 3292, 340, 9, 52, "Text"], Cell[145861, 3303, 331, 7, 108, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[146241, 3316, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[146455, 3323, 141, 2, 75, "Section"], Cell[146599, 3327, 94, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[146730, 3333, 141, 2, 55, "Section"], Cell[146874, 3337, 7515, 126, 1324, "Input"], Cell[154392, 3465, 107, 3, 46, "Text"], Cell[154502, 3470, 4694, 85, 778, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[159233, 3560, 141, 2, 55, "Section"], Cell[159377, 3564, 2040, 42, 401, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[161466, 3612, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[161680, 3619, 140, 2, 75, "Section"], Cell[161823, 3623, 93, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[161953, 3629, 141, 2, 55, "Section"], Cell[162097, 3633, 1258, 22, 386, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[163392, 3660, 141, 2, 55, "Section"], Cell[163536, 3664, 2897, 51, 624, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[166470, 3720, 139, 2, 55, "Section"], Cell[166612, 3724, 322, 6, 100, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[166971, 3735, 151, 2, 55, "Section"], Cell[167125, 3739, 126, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[167288, 3745, 141, 2, 55, "Section"], Cell[167432, 3749, 94, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[167563, 3755, 141, 2, 55, "Section"], Cell[167707, 3759, 94, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[167838, 3765, 140, 2, 55, "Section"], Cell[167981, 3769, 216, 6, 49, "Text"], Cell[168200, 3777, 1847, 37, 366, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[170084, 3819, 140, 2, 55, "Section"], Cell[170227, 3823, 1121, 35, 49, "Text"], Cell[171351, 3860, 700, 13, 192, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[172088, 3878, 141, 2, 55, "Section"], Cell[172232, 3882, 94, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[172363, 3888, 140, 2, 55, "Section"], Cell[172506, 3892, 3737, 67, 756, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[176280, 3964, 141, 2, 55, "Section"], Cell[176424, 3968, 94, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[176555, 3974, 143, 2, 55, "Section"], Cell[176701, 3978, 2775, 53, 586, "Input"], Cell[179479, 4033, 205, 3, 49, "Text"], Cell[179687, 4038, 1536, 26, 464, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[181260, 4069, 140, 2, 55, "Section"], Cell[181403, 4073, 377, 7, 117, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[181817, 4085, 140, 2, 55, "Section"], Cell[181960, 4089, 301, 7, 49, "Text"], Cell[182264, 4098, 1411, 28, 344, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[183712, 4131, 140, 2, 55, "Section"], Cell[183855, 4135, 575, 9, 190, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[184467, 4149, 140, 2, 55, "Section"], Cell[184610, 4153, 461, 8, 149, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[185108, 4166, 140, 2, 55, "Section"], Cell[185251, 4170, 204, 3, 82, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[185492, 4178, 140, 2, 55, "Section"], Cell[185635, 4182, 1055, 20, 280, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[186727, 4207, 140, 2, 55, "Section"], Cell[186870, 4211, 464, 10, 163, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[187371, 4226, 140, 2, 55, "Section"], Cell[187514, 4230, 499, 10, 195, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[188050, 4245, 159, 2, 55, "Section"], Cell[188212, 4249, 1168, 23, 288, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[189429, 4278, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[189643, 4285, 140, 2, 75, "Section"], Cell[189786, 4289, 201, 4, 79, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[190024, 4298, 140, 2, 75, "Section"], Cell[190167, 4302, 209, 4, 66, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[190413, 4311, 140, 2, 75, "Section"], Cell[190556, 4315, 248, 6, 66, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[190841, 4326, 140, 2, 75, "Section"], Cell[190984, 4330, 3867, 70, 526, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[194888, 4405, 166, 2, 75, "Section"], Cell[195057, 4409, 94, 1, 49, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[195188, 4415, 214, 5, 75, "Section"], Cell[195405, 4422, 760, 21, 49, "Text"], Cell[196168, 4445, 758, 13, 206, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[196963, 4463, 184, 2, 75, "Section"], Cell[197150, 4467, 477, 10, 167, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[197664, 4482, 183, 2, 75, "Section"], Cell[197850, 4486, 410, 8, 99, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[198297, 4499, 141, 2, 75, "Section"], Cell[198441, 4503, 1176, 23, 323, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[199654, 4531, 140, 2, 75, "Section"], Cell[199797, 4535, 99, 1, 49, "Text"], Cell[199899, 4538, 516, 9, 146, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[200452, 4552, 153, 2, 75, "Section"], Cell[200608, 4556, 2067, 42, 407, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[202712, 4603, 162, 2, 75, "Section"], Cell[202877, 4607, 2694, 54, 430, "Input"], Cell[205574, 4663, 254, 6, 49, "Text"], Cell[205831, 4671, 2826, 52, 466, "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[208706, 4729, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[208920, 4736, 142, 2, 75, "Section"], Cell[209065, 4740, 417, 10, 49, "Text"], Cell[209485, 4752, 3498, 69, 662, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[213020, 4826, 154, 2, 55, "Section"], Cell[213177, 4830, 735, 12, 206, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[213949, 4847, 141, 2, 55, "Section"], Cell[214093, 4851, 3696, 67, 626, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[217826, 4923, 169, 2, 55, "Section"], Cell[217998, 4927, 5648, 100, 950, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[223683, 5032, 166, 2, 55, "Section"], Cell[223852, 5036, 825, 14, 161, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[224714, 5055, 143, 2, 55, "Section"], Cell[224860, 5059, 1707, 33, 387, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[226604, 5097, 141, 2, 55, "Section"], Cell[226748, 5101, 2360, 45, 348, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[229145, 5151, 154, 2, 55, "Section"], Cell[229302, 5155, 225, 7, 49, "Text"], Cell[229530, 5164, 2605, 51, 739, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[232172, 5220, 181, 2, 55, "Section"], Cell[232356, 5224, 1141, 21, 186, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[233534, 5250, 649, 19, 55, "Section"], Cell[234186, 5271, 3693, 69, 626, "Input"], Cell[237882, 5342, 1246, 35, 49, "Text"], Cell[239131, 5379, 3249, 61, 678, "Input"], Cell[242383, 5442, 351, 5, 49, "Text"], Cell[242737, 5449, 1692, 32, 598, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[244466, 5486, 199, 5, 55, "Section"], Cell[244668, 5493, 1461, 28, 346, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[246166, 5526, 169, 2, 55, "Section"], Cell[246338, 5530, 3550, 65, 486, "Input"], Cell[249891, 5597, 350, 8, 87, "Text"], Cell[250244, 5607, 2843, 52, 486, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[253124, 5664, 162, 2, 55, "Section"], Cell[253289, 5668, 6158, 101, 1270, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[259484, 5774, 164, 2, 55, "Section"], Cell[259651, 5778, 1457, 25, 206, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[261157, 5809, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[261371, 5816, 138, 2, 75, "Section"], Cell[261512, 5820, 1190, 20, 287, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[262739, 5845, 138, 2, 55, "Section"], Cell[262880, 5849, 4796, 91, 842, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[267713, 5945, 138, 2, 55, "Section"], Cell[267854, 5949, 1841, 36, 386, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[269732, 5990, 138, 2, 55, "Section"], Cell[269873, 5994, 93, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[270003, 6000, 139, 2, 55, "Section"], Cell[270145, 6004, 84, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[270266, 6010, 156, 2, 55, "Section"], Cell[270425, 6014, 2443, 47, 487, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[272905, 6066, 177, 2, 55, "Section"], Cell[273085, 6070, 2409, 46, 286, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[275531, 6121, 178, 2, 55, "Section"], Cell[275712, 6125, 2148, 42, 306, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[277897, 6172, 166, 2, 55, "Section"], Cell[278066, 6176, 2455, 41, 326, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[280558, 6222, 160, 2, 55, "Section"], Cell[280721, 6226, 85, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[280843, 6232, 1086, 32, 55, "Section"], Cell[281932, 6266, 1499, 29, 466, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[283468, 6300, 165, 2, 55, "Section"], Cell[283636, 6304, 156, 4, 49, "Text"], Cell[283795, 6310, 9535, 197, 2126, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[293367, 6512, 181, 2, 55, "Section"], Cell[293551, 6516, 2571, 49, 491, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[296159, 6570, 167, 2, 55, "Section"], Cell[296329, 6574, 4501, 89, 906, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[300867, 6668, 186, 2, 55, "Section"], Cell[301056, 6672, 4494, 86, 1078, "Input"], Cell[305553, 6760, 1458, 30, 226, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[307048, 6795, 182, 2, 55, "Section"], Cell[307233, 6799, 84, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[307354, 6805, 166, 2, 55, "Section"], Cell[307523, 6809, 547, 9, 106, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[308107, 6823, 166, 2, 55, "Section"], Cell[308276, 6827, 6749, 113, 1238, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[315062, 6945, 209, 5, 55, "Section"], Cell[315274, 6952, 451, 9, 86, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[315762, 6966, 156, 2, 55, "Section"], Cell[315921, 6970, 1967, 39, 346, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[317925, 7014, 168, 2, 55, "Section"], Cell[318096, 7018, 961, 18, 166, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[319094, 7041, 167, 2, 55, "Section"], Cell[319264, 7045, 1595, 29, 266, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[320896, 7079, 139, 2, 55, "Section"], Cell[321038, 7083, 94, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[321169, 7089, 214, 5, 55, "Section"], Cell[321386, 7096, 1643, 31, 266, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[323066, 7132, 169, 2, 55, "Section"], Cell[323238, 7136, 1720, 35, 386, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[324995, 7176, 171, 2, 55, "Section"], Cell[325169, 7180, 262, 6, 49, "Text"], Cell[325434, 7188, 2904, 57, 566, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[328375, 7250, 680, 20, 55, "Section"], Cell[329058, 7272, 4785, 86, 1269, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[333880, 7363, 227, 5, 55, "Section"], Cell[334110, 7370, 325, 7, 49, "Text"], Cell[334438, 7379, 103, 1, 49, "Text"], Cell[334544, 7382, 8580, 169, 2506, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[343173, 7557, 182, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[343380, 7564, 144, 2, 75, "Section"], Cell[343527, 7568, 480, 10, 49, "Text"], Cell[344010, 7580, 1459, 26, 306, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[345506, 7611, 166, 2, 55, "Section"], Cell[345675, 7615, 2072, 36, 286, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[347784, 7656, 206, 5, 55, "Section"], Cell[347993, 7663, 10075, 181, 1847, "Input"], Cell[358071, 7846, 1740, 36, 326, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[359860, 7888, 186, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[360071, 7895, 164, 5, 102, "Section"], Cell[360238, 7902, 2661, 41, 366, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[362936, 7948, 145, 2, 55, "Section"], Cell[363084, 7952, 1448, 23, 336, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[364569, 7980, 139, 2, 55, "Section"], Cell[364711, 7984, 1320, 24, 296, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[366068, 8013, 162, 5, 82, "Section"], Cell[366233, 8020, 2808, 59, 446, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[369078, 8084, 201, 5, 82, "Section"], Cell[369282, 8091, 5645, 110, 766, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[374964, 8206, 164, 2, 75, "Section"], Cell[375131, 8210, 598, 13, 166, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[375766, 8228, 170, 2, 55, "Section"], Cell[375939, 8232, 2551, 49, 346, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[378527, 8286, 174, 2, 55, "Section"], Cell[378704, 8290, 3227, 65, 406, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[381968, 8360, 161, 5, 82, "Section"], Cell[382132, 8367, 2871, 49, 366, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[385040, 8421, 176, 5, 82, "Section"], Cell[385219, 8428, 2565, 49, 366, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[387821, 8482, 167, 5, 82, "Section"], Cell[387991, 8489, 213, 3, 49, "Text"], Cell[388207, 8494, 3974, 73, 546, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[392218, 8572, 429, 14, 56, "Section"], Cell[392650, 8588, 313, 6, 99, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[393000, 8599, 141, 2, 55, "Section"], Cell[393144, 8603, 619, 12, 146, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[393812, 8621, 186, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[394023, 8628, 179, 2, 75, "Section"], Cell[394205, 8632, 3108, 56, 426, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[397350, 8693, 140, 2, 55, "Section"], Cell[397493, 8697, 1128, 30, 68, "Text"], Cell[398624, 8729, 5311, 96, 730, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[403972, 8830, 153, 2, 55, "Section"], Cell[404128, 8834, 2410, 46, 546, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[406575, 8885, 148, 2, 55, "Section"], Cell[406726, 8889, 1700, 33, 346, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[408463, 8927, 151, 2, 55, "Section"], Cell[408617, 8931, 864, 18, 191, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[409518, 8954, 178, 2, 55, "Section"], Cell[409699, 8958, 10392, 205, 1603, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[420128, 9168, 140, 2, 55, "Section"], Cell[420271, 9172, 1095, 22, 226, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[421403, 9199, 174, 2, 55, "Section"], Cell[421580, 9203, 1652, 31, 295, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[423269, 9239, 163, 2, 55, "Section"], Cell[423435, 9243, 889, 18, 149, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[424361, 9266, 170, 2, 55, "Section"], Cell[424534, 9270, 1029, 20, 148, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[425600, 9295, 140, 2, 55, "Section"], Cell[425743, 9299, 4589, 87, 606, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[430369, 9391, 163, 2, 55, "Section"], Cell[430535, 9395, 3624, 76, 906, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[434196, 9476, 142, 2, 55, "Section"], Cell[434341, 9480, 2483, 51, 446, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[436873, 9537, 186, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[437084, 9544, 138, 2, 75, "Section"], Cell[437225, 9548, 3409, 68, 626, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[440671, 9621, 140, 2, 75, "Section"], Cell[440814, 9625, 1047, 19, 240, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[441898, 9649, 138, 2, 75, "Section"], Cell[442039, 9653, 662, 11, 206, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[442738, 9669, 142, 2, 75, "Section"], Cell[442883, 9673, 1778, 35, 406, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[444698, 9713, 139, 2, 55, "Section"], Cell[444840, 9717, 562, 11, 206, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[445439, 9733, 138, 2, 55, "Section"], Cell[445580, 9737, 2749, 54, 586, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[448366, 9796, 158, 2, 55, "Section"], Cell[448527, 9800, 10794, 211, 1586, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[459358, 10016, 183, 5, 82, "Section"], Cell[459544, 10023, 1874, 37, 386, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[461455, 10065, 137, 2, 55, "Section"], Cell[461595, 10069, 3100, 63, 747, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[464732, 10137, 160, 2, 55, "Section"], Cell[464895, 10141, 4508, 89, 766, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[469440, 10235, 139, 2, 55, "Section"], Cell[469582, 10239, 2310, 46, 594, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[471929, 10290, 161, 5, 82, "Section"], Cell[472093, 10297, 913, 16, 146, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[473055, 10319, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[473269, 10326, 140, 2, 75, "Section"], Cell[473412, 10330, 138, 2, 81, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[473587, 10337, 140, 2, 55, "Section"], Cell[473730, 10341, 1641, 34, 246, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[475408, 10380, 140, 2, 55, "Section"], Cell[475551, 10384, 1015, 18, 166, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[476603, 10407, 138, 2, 55, "Section"], Cell[476744, 10411, 2573, 54, 506, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[479354, 10470, 138, 2, 55, "Section"], Cell[479495, 10474, 190, 4, 66, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[479722, 10483, 138, 2, 55, "Section"], Cell[479863, 10487, 247, 5, 66, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[480147, 10497, 138, 2, 55, "Section"], Cell[480288, 10501, 169, 5, 49, "Text"], Cell[480460, 10508, 1594, 30, 406, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[482091, 10543, 139, 2, 55, "Section"], Cell[482233, 10547, 329, 6, 66, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[482599, 10558, 144, 2, 55, "Section"], Cell[482746, 10562, 222, 4, 66, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[483005, 10571, 144, 2, 55, "Section"], Cell[483152, 10575, 221, 6, 49, "Text"], Cell[483376, 10583, 1358, 29, 186, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[484771, 10617, 144, 2, 55, "Section"], Cell[484918, 10621, 412, 9, 86, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[485367, 10635, 141, 2, 55, "Section"], Cell[485511, 10639, 272, 7, 49, "Text"], Cell[485786, 10648, 371, 7, 66, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[486194, 10660, 139, 2, 55, "Section"], Cell[486336, 10664, 281, 8, 49, "Text"], Cell[486620, 10674, 658, 14, 106, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[487315, 10693, 139, 2, 55, "Section"], Cell[487457, 10697, 174, 5, 49, "Text"], Cell[487634, 10704, 302, 6, 66, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[487973, 10715, 141, 2, 55, "Section"], Cell[488117, 10719, 94, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[488248, 10725, 158, 2, 55, "Section"], Cell[488409, 10729, 2841, 48, 546, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[491287, 10782, 175, 2, 55, "Section"], Cell[491465, 10786, 2014, 44, 526, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[493516, 10835, 139, 2, 55, "Section"], Cell[493658, 10839, 454, 11, 49, "Text"], Cell[494115, 10852, 1366, 31, 326, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[495518, 10888, 141, 2, 55, "Section"], Cell[495662, 10892, 635, 11, 126, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[496334, 10908, 139, 2, 55, "Section"], Cell[496476, 10912, 158, 2, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[496671, 10919, 178, 2, 55, "Section"], Cell[496852, 10923, 348, 8, 49, "Text"], Cell[497203, 10933, 7090, 141, 1066, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[504330, 11079, 139, 2, 55, "Section"], Cell[504472, 11083, 527, 12, 68, "Text"], Cell[505002, 11097, 3470, 74, 686, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[508509, 11176, 139, 2, 55, "Section"], Cell[508651, 11180, 537, 12, 118, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[509225, 11197, 183, 2, 55, "Section"], Cell[509411, 11201, 307, 8, 49, "Text"], Cell[509721, 11211, 3536, 69, 746, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[513294, 11285, 210, 5, 55, "Section"], Cell[513507, 11292, 158, 2, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[513702, 11299, 164, 2, 55, "Section"], Cell[513869, 11303, 1625, 31, 346, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[515531, 11339, 139, 2, 55, "Section"], Cell[515673, 11343, 334, 8, 49, "Text"], Cell[516010, 11353, 3764, 73, 946, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[519811, 11431, 175, 2, 55, "Section"], Cell[519989, 11435, 212, 4, 73, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[520238, 11444, 208, 5, 55, "Section"], Cell[520449, 11451, 749, 14, 232, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[521247, 11471, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[521461, 11478, 138, 2, 75, "Section"], Cell[521602, 11482, 183, 5, 49, "Text"], Cell[521788, 11489, 519, 12, 206, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[522344, 11506, 288, 7, 57, "Section"], Cell[522635, 11515, 1405, 36, 68, "Text"], Cell[524043, 11553, 2577, 53, 875, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[526657, 11611, 163, 2, 55, "Section"], Cell[526823, 11615, 99, 1, 49, "Text"], Cell[526925, 11618, 1572, 29, 331, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[528534, 11652, 141, 2, 55, "Section"], Cell[528678, 11656, 2111, 44, 382, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[530826, 11705, 164, 2, 55, "Section"], Cell[530993, 11709, 172, 4, 49, "Text"], Cell[531168, 11715, 5978, 103, 1230, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[537183, 11823, 198, 5, 55, "Section"], Cell[537384, 11830, 1245, 26, 368, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[538666, 11861, 878, 27, 56, "Section"], Cell[539547, 11890, 95, 1, 49, "Text"], Cell[539645, 11893, 705, 11, 140, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[540387, 11909, 139, 2, 55, "Section"], Cell[540529, 11913, 158, 2, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[540724, 11920, 139, 2, 55, "Section"], Cell[540866, 11924, 158, 2, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[541061, 11931, 260, 5, 55, "Section"], Cell[541324, 11938, 95, 1, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[541456, 11944, 205, 5, 55, "Section"], Cell[541664, 11951, 1508, 30, 255, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[543221, 11987, 189, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[543435, 11994, 141, 2, 75, "Section"], Cell[543579, 11998, 2318, 46, 469, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[545934, 12049, 166, 2, 55, "Section"], Cell[546103, 12053, 162, 2, 49, "Text"], Cell[546268, 12057, 6849, 133, 1668, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[553154, 12195, 141, 2, 55, "Section"], Cell[553298, 12199, 3004, 57, 548, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[556339, 12261, 141, 2, 55, "Section"], Cell[556483, 12265, 2758, 54, 620, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[559290, 12325, 182, 3, 71, "Subtitle"], Cell[CellGroupData[{ Cell[559497, 12332, 169, 2, 75, "Section"], Cell[559669, 12336, 3187, 65, 509, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[562893, 12406, 207, 5, 75, "Section"], Cell[563103, 12413, 9449, 179, 1596, "Input"], Cell[572555, 12594, 4736, 89, 692, "Input"], Cell[577294, 12685, 2047, 40, 350, "Input"], Cell[579344, 12727, 2880, 53, 525, "Input"], Cell[582227, 12782, 2628, 49, 478, "Input"], Cell[584858, 12833, 97, 2, 66, "Input"] }, Open ]] }, Closed]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)