Math 642 Midterm Review – Spring 2010

Green's functions & unbounded operators

• Section 4.2
  1. Be able to find Green's functions for 2nd order ODEs
• Section 4.3
  1. Unbounded operators and their domains
  2. Adjoints of unbounded operators (2nd order)
  3. Self-adjoint operators and eigenvalue problems

Calculus of variations

• Section 5.1
  1. Know the difference between Frechet and Gateaux derivatives.
  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. Be able to sketch a proof of the coordinate invariance for the Euler-Lagrange equations. (See class notes (2/1/2010) for this.)

• Section 5.2
  1. Hamilton's principle; Lagrangians
  2. Legendre transformations and Hamiltonians
  3. 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.
  4. Lagrangian for a stretched, vibrating string, or other similar problems.

• 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. Rayleigh-Ritz principle
  3. Courant-Fischer minimax theorem

Complex analysis

• Section 6.1
  1. Cauchy-Riemann equations and analytic functions
  2. Branch cuts and branch points
• Sections 6.2 & 6.4 and Residues and Contour Integration Problems
  1. Cauchy's theorem, Cauchy's integral formulas
  2. Taylor and Laurent series
  3. Classification of isolated singularities: removable, pole of order m, essential
  4. Residues and the residue theorem
  5. Evaluation of real integrals via contour integration, with and without cuts

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 3/3/2010 (fjn).