Johan A.C. Kolk

Courses at University College Utrecht

2007 Spring: UCU Science 311: Advanced Mathematics I

Instructors

Lecturer: Johan A.C. Kolk (e-mail: kolk@math.uu.nl).
Personal Homepage of lecturer: http://www.math.uu.nl/people/kolk, in particular, the page Links contains many useful resources);
Teaching Assistant: Jakub Byszewski (e-mail: byszewsk@math.uu.nl).

Time and place

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.

Textbook

Gilbert Strang: Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, 1986.
For more information, see Wellesley-Cambridge Press as well as the home page for the course at MIT (= Massachusetts Institute of Technology, Cambridge, MA, USA) as recently taught by the author. This page is a rich source of extra information: hand-outs for the lectures, class resources, old exams, etc. Currently Professor Strang is president of SIAM (= Society for Industrial and Applied Mathematics).

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

Mathematica

Using the computer program Mathematica in illustrative examples is an important part of the course. The programme enables one to bypass tedious computations and to concentrate on the mathematical interpretation right away. Furthermore, some mastery of such a tool is a standard prerequisite in many master's programmes, while the industry definitely assumes that it belongs to the toolbox of its prospective employees active in the (hard) sciences. Mathematica has an extensive help function. Very illuminating are the "Tour" and the "Demos": one is strongly advised to spend some time studying (some parts of) these. Useful is the quick reference sheet.

Mathematica notebook containing solutions to some exercises in the book.

This can be downloaded here.

Mathematica notebook used in the computer sessions

This can be downloaded here.

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

Amico.

Manual

An outdated manual can be downloaded here.

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.

In particular, therefore, there will be neither a written midterm nor a final exam.
The final grade will be determined by the following algorithm: the four ingredients count for 25, 25, 20 and 30 percent, respectively.
In determining the grade, UCU's rules about class attendance will be applied. In particular, missing more than 25 percent of all classes will automatically result in the grade F, unless in case of circumstances beyond one's control.

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 22, 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.
- Jan 25, 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 2
- Jan 29, No lecture, because of the lecturer being abroad.
- Feb 01, Computer session: ON DEMAND there will be one, in order to enable students to improve their proficiency in Mathematica. In that case, the "Tour" and the "Demos" can be studied, while the teaching assistant will be available.
- Homework: None.
Week 3
- Feb 05, Lecture: Remainder of Minimum Principles, Section 1.4.
- Feb 08, 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 12, 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 15, Computer session.
- Homework:
  - Section A Framework for the Applications and Min-Max Duality in the Mathematica notebook (download the updated version). The latter part 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 19, 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 visual information about the first application of Lagrange multipliers, see the following Mathematica notebook. In addition, for a more thorough, mathematical, discusssion of min-max duality see here; this has been an examination question in the course WISB 212. For optional information about the Rayleigh quotient, see this Mathematica notebook.
- Feb 22, Computer session.
- Homework:
  - Subsection Electrical Networks in the Mathematica notebook (download the new version). This is concerned with applications of known theory.
  - Exercises 2.2.4, 2.2.7 and part 1 of 2.3.4.
Week 6
- Feb 26, Lecture: Structures in Equilibrium, Section 2.4 until Interpretation of Lagrange Multipliers.
- Mar 01, Computer session: The Subsection Trusses: Part I from the Mathematica notebook (download the new version). This is concerned with applications of known theory.
- Homework:
  - Exercises 2.4.2 (number the nodes clockwise starting from the upper left node and the bars in the order given by the pairs of nodes (1,4), (1,2), (2,3), (1,3) and (2,4)), 2.4.3, 2.4.16.
- 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 Soda Straw Tensegrity Structures by G.W. Hart.
Week 7
- Mar 05, Lecture: Virtual Work from Section 2.4, start on Equilibrium in the Continuous Case from Section 3.1.
- Mar 08, Computer session: The Subsection Trusses: Part II in the Mathematica notebook (download the new version). This is concerned with applications of known theory.
- The Mathematica notebook contains the subsection Midterm Project. This part of the notebook contains problems that jointly constitute the midterm project mentioned above under Grade. The ultimate time of electronic submission of project is Thursday, March 22 at 14:00 hour. In particular, there will be no midterm examination.
Week 8
- Mar 12, Lecture: The Hanging Bar and Sturm-Liouville Problems from Section 3.1.
- Mar 15, Computer session: Subsection Sturm-Liouville from the Mathematica notebook (download the new version).
- Homework:
  - Exercises 3.1.1, 3.1.2, 3.1.6, 3.1.10.
Week 8 bis
- Mar 19 and 22, Spring Break: No Classes.
Week 9
- Mar 26, 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.
- Mar 29, Computer session: Start on the Subsection Soap Bubbles in the Plane 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 02, 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 05, Computer session: Subsection Galerkin Method from the Mathematica notebook (download the new version).
- Homework:
  - Subsection Galerkin Method in the Mathematica notebook.
  - Exercises 3.6.5, 3.6.6, 3.2.12.
Week 11
- Apr 09, Easter Monday: No lecture.
- Apr 12, Lecture: The end of Section 3.2 starting from the Subsection Fourth-order Equations.
- For some of the computations concerning the Hermite cubic and splines, see this Mathematica notebook.
- For a mathematical treatment of basic splines, see Exercise 6.109 in: J.J. Duistermaat and J.A.C. Kolk, Multidimensional Real Analysis, Vol II, Cambridge University Press, 2004; and for distributions, see J.J. Duistermaat and J.A.C. Kolk, Distributions: Theory and Applications, Birkhäuser Verlag, Boston, to appear.
- An additional resource: Beams, Bending, and Boundary Conditions.
Week 12
- Apr 16, Computer session: Play with the spline and the Bezier curve applets of Mark Hoefer.
- Work through the Subsections Interpolation: Displacements and Slopes and Cubic Splines, Bezier Curves and Calculus of Variations from the Mathematica notebook (download the new version). Solve the one exercise occurring in the last subsection.
- Homework:
  - Exercises 3.2.10, 3.2.13, 3.2.17.
- Apr 19, 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
- Apr 23, 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.
- Apr 26, 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
- Apr 30, Queen's Day: no lecture.
- May 03, Computer session: an occasion to put your questions on the soap project to the instructors.
- 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.
Week 15
- May 07 and 10, No classes: Exam Week.
- YOUR PROJECT IS DUE ULTIMATELY MONDAY, MAY 14, 12:00 noon, 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, Budapestlaan 6, 3584 CD Utrecht. 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: Feb 21, 2007
J.A.C. Kolk