Math 642 Midterm Review — Spring 2017

The midterm for Math 642 will be held on Wednesday, March 22. The test covers these sections from the text: 5.1, 5.2 (except 5.2.3-5.2.5), 5.4, 6.2.3, 6.2.4, 6.4, and 6.5.1.

Calculus of Variations

• Section 5.1
1. Frechet and Gateaux derivatives, definitions.
2. Be able to derive the Euler-Lagrange equations, using variational calculations, for constrained and unconstrained problems that are subject to various boundary conditions.
3. Coordinate invariance of an extremal under change of coordinates. Geodesics.

• Section 5.2
1. Hamilton's principle; Lagrangians.
2. Legendre transformations and Hamiltonians.
3. Hamilton's equations; conserved quantities — angular momentum, energy.
4. Be able to derive Lagrangians and Hamiltonians for simple mechanical systems, such as a mass subject to a central force (radial potential) or a pendulum.

• Section 5.4
1. Be able to use variation methods for finding eigenvalues and eigenfunctions in a Sturm-Liouville problem, subject to various boundary conditions.
2. Minimum principle.
3. Courant-Fischer minimax theorem; be able to sketch a proof.

Complex variables

• Section 6.2
1. Cauchy's theorem, Cauchy's integral formulae.
2. Taylor and Laurent series.
3. Isolated singularities; classification: removable; pole of order $m$; essential.
4. Residues and the residue theorem.
5. Isolation of zeros. Be able to prove the zeros of an analytic function are isolated. Be able to use this to prove various identities hold.

• Section 6.4
1. Jordan's lemma. Be able to apply it and be able to prove it.
2. Contour integration, with and without cuts.

• Section 6.5
1. Gamma function
• Definition and extension to $\mathbb C \setminus \{0,-1,-2,\cdots\}$
• Fundamental identity (Keener, Eqn. 6.25). Be able to sketch a proof.)

Structure of the exam

The test will have two parts. The in-class part will have 4 or 5 questions. You will be asked to state definitions and to do problems similar to assigned homework problems and examples done in class. In addition, you will be asked prove or sketch a proof for a major theorem or lemma from the material covered by this test. The take-home part will consist of computations and proofs of theorems.

Updated 3/14/17 (fjn).