Johan A.C. Kolk

The lectures take place on Monday, 16:00-18:00 and the computer sessions on Thursday, 8:45-10:45 both in the Newton Building, Lecture room F.

The file Addenda and corrigenda contains material additional to this textbook.

Mathematica

Mathematica notebook containing solutions to some exercises in the book.

Mathematica notebook used in the computer sessions

Homework

Solutions to the exercises from the book that are assigned during one week, are to be handed in to the lecturer at the start of the lecture on the next Monday.
Similarly, solutions to the questions in the Mathematica notebook are to be sent by e-mail to the teaching assistant before the next Wednesday, 1:00 pm.
Late homework will not be accepted, unless with prior notification or in case of circumstances beyond one's control. During the course, it will be also the students responsibility to make sure that assignments are returned to them in due time: late claims will make a weak case.

Amico Workspaces

Manual

An outdated manual can be downloaded here.

Mailing list

Mailing list of students participating in the course.

Grade

Four ingredients will be used in determining the grade for the course:

• written homework, which directly illustrates the theory in simple situations;
• submitted Mathematica notebooks, which consider more realistic examples;
• midterm project, consisting of some applications to be studied mainly using Mathematica;
• final project, consisting of application of some Mathematica routines, which serve as a starting point, and of a mathematical discussion of their consequences; furthermore, original contributions are encouraged.

Schedule

This page will be updated regularly. Here you will find a tentative schedule of the activities for the course, listed by week.

• Week 1
  • Jan 30, Lecture: No lecture, because of the lecturer being abroad.
  • Feb 02, Lecture: None, for the same reason as above.
  • Homework: None.
  
  
• Week 2
  • Feb 06, Lecture: Symmetric Matrices, Positive Definiteness, study Sections 1.1 and 1.2 and the initial part of Section 1.3, until The Minimum of a Quadratic.
  • Feb 09, Lecture: Completing Squares, Sections 1.2 and 1.3 and Minimum Principles, Section 1.4, until Example 1 on page 35.
  • Homework:
    • Exercises 1.2.6, 1.2.7, 1.2.8, 1.3.1, 1.3.5.
  
  
• Week 3
  • Feb 13, Lecture: Remainder of Minimum Principles, Section 1.4.
  • Feb 16, Computer session: Mathematica notebook 311. The Section Least Squares of the notebook is to be handed in before the next practice session.
  • Homework:
    • Exercises 1.3.16, 1.4.2, 1.4.9, 1.5.3.
  
  
• Week 4
  • Feb 20, Lecture: From Section 1.5 the Subsections Diagonal Form and the spectral theory for Symmetric Matrices, starting on page 60. Furthermore A Framework for the Applications in Section 2.1.
  • Feb 23, Computer session.
  • Homework:
    • Section Min-Max Duality in the Mathematica notebook. This constitutes a preparation for the upcoming lecture.
    • Exercises 1.5.7, 1.5.22, 1.5.23. For the last two exercises you may use Mathematica to find eigenvalues and eigenvectors, in particular the routine Eigensystem, if you wish. What is the condition on a vector x that is needed to turn the matrix xx^T into a projection? The package <<LinearAlgebra`Orthogonalization` in Mathematica is useful when scaling vectors to unit length. The routine Map may be used to apply Normalize to several vectors simultaneously. When desperate, look here.
  
  
• Week 5
  • Feb 27, Lecture: Constraints and Lagrange Multipliers, Section 2.2 up to Projections on page 105 and Electrical Networks, Section 2.3 until RLC Circuits on page 115. For optional information about the Rayleigh quotient, see the following Mathematica notebook.
  • Mar 02, Computer session.
  • Homework:
    • Subsection Networks in the Mathematica notebook. This is concerned with applications of known theory.
    • A network is called a tree if one can reach every node from every other, but there is no loop. A tree is called rooted if one of the nodes has been designated "the root". Let us call a rooted tree planted if its root has been grounded. Consider a planted tree T with n edges. How many nodes does it have, apart from the root? Let A be the edge-node incidence matrix from which the column of the root has been deleted. If there are no current sources at the nodes, what can be said about the currents in the tree? What does this imply about the rank of A? And if there are no voltage sources along the edges, what conditions must the voltages at the nodes satisfy to avoid currents in T? What does that say about the rank of A? Your comment?
    • Exercise 2.2.4.
  
  
• Week 6
  • Mar 06, Lecture: Impedances from Section 2.3 and Structures in Equilibrium, Section 2.4 until Interpretation of Lagrange Multipliers.
  • Mar 09, Computer session: Subsection Impedances and start on the Subsection Trusses from the Mathematica notebook.
  • Homework:
    • This week the Subsection Trusses is part of a larger project, so you do not have to hand this in next week, but the week after. Impedances and the other exercises are to be handed in at the usual time.
    • Exercises 2.2.2, 2.4.13.
    • Exercise 2.4.16, which in my copy reads:
      Suppose a truss consists of one bar at an angle theta with the horizontal. Sketch forces f₁ and f₂ at the upper end, acting in the positive x and y directions, and corresponding forces f₃ and f₄ at the lower end. Write down the 1 by 4 matrix A₀, the 4 by 1 matrix A₀^T, and the 4 by 4 matrix A₀^T C A₀. For which forces can the equation A₀^T y = f be solved?
    • For all you always wanted to know about trusses, see Analysis of Structural Member Systems.
    • For the programme Bridge Designer about trusses developed by the School of Engineering in Johns Hopkins University, see the Virtual Laboratories.
    • For applications of the theory in computer games, see Chronic Logic or BridgeBuilder.
    • Sometimes things may really go wrong, see Engineering at Carleton University.
    • For building tensegrity structures, see here.
  
  
• Week 7
  • Mar 13, Lecture: Virtual Work from Section 2.4, start on Equilibrium in the Continuous Case from Section 3.1.
  • Mar 16, Computer session: Finish the Subsection Trusses in the Mathematica notebook.
  • Homework:
    • Subsection Trusses in the Mathematica notebook.
    • Investigate in the case of equal vertical loads what is significant for the amount of bending. Is it the stiffness of the horizontal bars, the non-horizontal bars, their ratio? (We assume all horizontal bars are equally stiff, and all non-horizontal bars are equally stiff.)
    • Back to the case where the horizontal bars have low stiffness. How should you choose the external forces at the free end of the truss to illustrate its bendability?
  
  
• Week 8
  • Mar 20, Lecture: The Hanging Bar and Sturm-Liouville Problems from Section 3.1.
  • Mar 23, Computer session: Subsection Sturm-Liouville from the Mathematica notebook.
  • Homework:
    • Exercises 3.1.1, 3.1.2, 3.1.6, 3.1.10.
  
  
• Week 8 bis
  • Mid-term break
  
  
• Week 9
  • Apr 03, Lecture: From Section 3.2 on Differential Equations of Equilibrium the Subsection Minimum Principles till Remark 1. From Section 3.6 on Calculus of Variations the Subsection One-dimensional Problems.
  • Apr 06 , Computer session: Start on the Subsection Soap Bubbles from the Mathematica notebook, in particular Fact 2. This is part of the term project, so you do not hand it in as homework. Eventually you incorporate it into your Term Project on Soap.
  • For a rich source of information on minimal surfaces, see Ken Brakke's Home Page.
  • And here are additional sources: Exploratorium.
  • Plateau's Problem.
  • Standard and Nonstandard Double Bubbles.
  • Some Results on Bubbles.
  • Homework:
    • Exercises 3.2.1, 3.2.2, 3.6.2.
  
  
• Week 10
  • Apr 10, Lecture: Remaining part of Section 3.6 Calculus of Variations until the subsection Nonlinear Equations. Beginning of treatment of numerical aspects, in particular, the finite element method from Remark 3 in Subsection Minimum Principles in Section 3.2 as well as the first subsection of Section 5.4 The Finite Element Method.
  • Apr 13, Computer session: Subsection Galerkin Method from the Mathematica notebook.
  • Homework:
    • Subsection Galerkin Method in the Mathematica notebook.
    • Exercises 3.6.5, 3.6.6, 3.2.12.
  
  
• Week 11
  • Apr 17, Easter Monday: No lecture.
    
  • Apr 20, Lecture: The end of Section 3.2 starting from the Subsection Fourth-order equations. For some of the computations about the Hermite cubic in the book, see this Mathematica notebook.
  
  
• Week 12
  • Apr 24, Computer session: Play with the Bezier curve and the spline applets of Mark Hoefer.
  • Work through the Mathematica notebook on Bezier curves and splines.
  • Work through the Mathematica notebook on minimal surfaces and solve the one exercise occurring in it.
  • Homework:
    • Exercises 3.2.10, 3.2.13, 3.2.17.
  • Apr 27, Lecture: Laplace's Equation and Potential Flow from Section 3.3 until page 195. The main concern of this part of the book is putting well-known results about vector analysis in the two-dimensional plane in the context developed so far.
    For Mathematica notebooks concerned with harmonic functions, which are produced by Sheldon Axler, see here.
  
  
• Week 13
  • May 01, Computer session: Work through the Mathematica notebook on the Laplace equation.
  • Homework:
    • Exercises 3.3.6, 3.3.10, 3.3.11.
    • Hand in Exercise 3.3.14 from the computer session.
  • May 04 , Lecture: Complex Variables and Conformal Mapping from Section 4.4 until page 346. Here we see a remarkable application of the theory of complex functions in obtaining solutions of the Laplace equation satisfying suitable boundary conditions. Take a look at Exercises 3.3.8, 3.3.14, 3.3.15, 3.3.17, 3.3.18. It shouldn't always be necessary to use a computer, but if so, you might find the following Mathematica notebook useful.
    
  • ALTERNATIVE PROJECT, instead of the Project on the Properties of Two-dimensional Soap. For more information, look here.
  
  
• Week 14
  • May 08, Computer session: an occasion to put your questions on the soap project to the lecturer.
  • Make sure you are not underestimating your project. You should have tried many things by now. To write a good story, you must understand things well: the global picture as well as the details. Furthermore, both contents and presentation of your paper should be well thought-out in order to get a good grade for the course. In particular, it must satisfy the customary requirements for a report, e.g., there should be a summary, a global description of the problem and of your method(s) of solution, and a transparant exposition of the mathematics you are using. In addition, it should be readable by a mathematician who doesn't know the field (of two-dimensional soap). And, by the way, it has to be original.
  • May 11, ON REQUEST, there will be an opportunity to discuss your problems in the project with the lecturers.
  
  
• Week 15
  • May 15, No Computer session: work on the project.
  • May 18, No Lecture: work on the project.
  • YOUR PROJECT IS DUE ULTIMATELY MONDAY, MAY 22, 12:00 AM, to be sent to the lecturer, by e-mail or ordinary mail, or hand-delivered in the mailbox at the ground floor of the Mathematical Institute. Make sure you have a copy yourself. Past experience has shown that only Mathematica notebooks or PDF files can be printed without problems; on the other hand, Micosoft Word files with embedded formulae or pictures may cause a lot of trouble under the UNIX system of the Mathematical Institute (more esoteric formats are out of the question anyway). As a consequence, only files of the former type may be submitted electronically and those of the latter kind in hard copy only.

Last modified: May 29, 2006
J.A.C. Kolk