Math 642 Midterm Review – Spring 2016

The midterm for Math 642 will be held on Wednesday, March 9. The test covers these sections from the text: 5.1, 5.2, 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 (see notes 2/24/16).
6. Maximum modulus theorem.

• 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 ${\mathbb C}\! \setminus \!{\mathbb N}_{\le 0}$
• 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 3/3/15 (fjn).