This can be downloaded here; the last up-date was October 15, 2004.

Mailing list

Grade

The grade for the course will be determined by the following algorithm: the homework, that is, exercises and work on computer notebooks, counts for two thirds and the project for one third.

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
  • Aug 31, Lecture: Symmetric matrices, positive definiteness, study at least Sections 1.1 and 1.2.
  • Sep 03, Lecture: Completing squares, Sections 1.2 and 1.3.
  • Homework:
    • Exercises 1.2.6, 1.2.7, 1.2.8, 1.3.1, 1.3.5.
      The homework is to be handed in to the teaching assistant during the next practice session, at the start of the meeting.
  
  
• Week 2
  • Sep 07, Lecture: Minimum principles, Section 1.4.
  • Sep 10, Computer session: Mathematica notebook 311. The Section Least squares of the notebook is to be handed in during the next practice session, so you may want to start it now. If you submit it electronically, follow these instructions.
  • Homework:
    • Exercises 1.3.16, 1.4.2, 1.4.9, 1.5.3.
  
  
• Week 3
  • Sep 14 Lecture: From Section 1.5 the Subsections Diagonal form and the spectral theory for Symmetric matrices. Furthermore Electrical network from Section 2.1.
  • Sep 17, Computer session.
  • Homework:
    • Section Duality in the Mathematica notebook.
    • 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 4
  • Sep 21, Lecture: Constraints and Lagrange multipliers, Section 2.2 up to duality. The structural equilibrium for trusses, Section 2.4.
  • Sep 24, Computer session.
  • Homework:
    • Subsections Networks and Impedances in the Mathematica notebook.
    • 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 5
  • Sep 28, Lecture: Duality  from Section 2.2 and Impedances from Section 2.3.
  • Oct 01, Computer session: Start on the Subsection Trusses from the Mathematica notebook.
  • Homework:
    • This week the notebook is part of a larger project, so you do not have to hand it in next week, but the week after. 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?
  
  
• Week 6
  • Oct 05, Lecture: Virtual work from Section 2.4, start on Equilibrium in the continuous case from Section 3.1.
  • Oct 08, 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 7
  • Oct 12, Lecture: The Hanging bar and Sturm-Liouville problems from Section 3.1.
  • Oct 15, Computer session: Subsection Sturm-Liouville from the Mathematica notebook.
  • Homework:
    • Exercises 3.1.1, 3.1.2, 3.1.6, 3.1.10.
  
  
• Week 7 bis
  • Fall break
  
  
• Week 8
  • Oct 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.
  • Oct 29, 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.
  • Homework:
    • Exercises 3.2.1, 3.2.2, 3.6.2.
  
  
• Week 9
  • Nov 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.
  • Nov 05, 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 10
  • Nov 09, 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.
  • Nov 12, Computer session: Play with the Bezier curve and the spline applets of Mark Hoefer.
  • Homework:
    • Exercises 3.2.10, 3.2.13, 3.2.17.
    • Work through the Mathematica notebook on Bezier curves and splines.
    • Get somewhere with Fact 2 of the Subsection Soap bubbles from the Mathematica notebook.
  
  
• Week 11
  • Nov 16, Lecture: Laplace's equation and potential flow from Section 3.3 until page 195.
  • Nov 19, Computer session: 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.
  • Homework:
    • Exercises 3.3.6, 3.3.10, 3.3.11.
    • Hand in Exercise 3.3.14 from the computer session.
    • Work through the Mathematica notebook on the Laplace equation.
  
  
• Week 12
  • Nov 23, Lecture: Two-dimensional Soap, see also the handout.
  • Nov 26, Computer session: the last occasion to put your questions on the soap project to the lecturers. Next week we will start new topics.
  • Make sure you are not underestimating the soap 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 13
  • Nov 30, 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.
  • Dec 03, Computer session: work on the Mathematica notebook on conformal mappings.
  • Homework:
    • Exercises 4.4.15, 4.4.16, 4.4.18.
  
  
• Week 14
  • Dec 07, No Lecture: work on the project.
  • Dec 10, No Computer session: work on the project.
  
  
• Week 15
  • Dec 14, No Lecture, in view of Examination week.
  • Dec 17, No Computer session.
  • Hand in Term Project on Soap ultimately by Dec 14.