Math 642 Midterm Review – Spring 2011
The midterm for Math 642 will be held on Friday, March 25. The test
will have an in-class part and a take-home part. The in-class part
covers sections 5.1, 5.2 & 5.4, and the take-home covers 6.1, 6.2,
6.4 & 6.5. Both parts include material discussed in pertinent
material discussed in class. Below is a review of topics to be covered
on the in-class part of the test.
Calculus of variations
- Section 5.1
- Know the difference between Frechet and Gateaux derivatives.
- Be able to derive the Euler-Lagrange equations, using variational
calculations, for constrained and unconstrained problems that are
subject to various boundary conditions -- Dirichlet, natural, etc.
- Be able to sove simple calculus of variations problems,
constrained or unconstrained.
- Section 5.2
- Hamilton's principle; Lagrangians.
- Legendre transformations and Hamiltonians
- Be able to derive equations of motion for simple mechanical
systems, such as a mass subject to a central force (radial potential)
or a pendulum. Doing these will require knowing the arclength
ds2 in polar and spherical coordinates, and making use of
extremals being invariant under a change of coordinates.
- Lagrangian for a stretched, vibrating string, or other similar
problems.
- Section 5.4
- Be able to use variation methods for finding eigenvalues and
eigenfunctions in a Sturm-Liouville problem, subject to various
boundary conditions -- Dirichlet, Neumann, mixed.
- Rayleigh-Ritz principle. Be able to state and prove this, as well
as use it.
- Courant-Fischer minimax theorem. Be able to state and prove this,
as well as use it.
Structure of the exam
There will be 4 to 6 questions. You will be asked to state a few
definitions, and to do problems similar
to
assigned homework problems and examples done in class. In
addition, you will be asked to give a derivation or a proof for a
major theorem or lemma from the material covered by this test.
Updated 3/21/2011 (fjn).