Math 642 Midterm Review – Spring 2007

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 consrtained 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 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 with several dependent variables.

• 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 variables and special functions

• Section 6.1
  1. Cauchy-Riemann equations
  2. Analytic function
  3. Branch cuts and branch points
• Section 6.2
  1. Cauchy's theorem, Cauchy's integral formula
  2. Taylor and Laurent series
  3. Residues and the residue theorem
  4. Know the maximum principle and the version of the Phragmen-Lindelof theorem in problem 14, section 6.2. (You just need to know these, not prove them.)
  5. Analytic continuation (monodromy theorem)
• Section 6.4
  1. Jordan's lemma
  2. Contour integration, with and without cuts
• Section 6.5
  1. Gamma function
  2. Beta function
  3. Bessel functions
     • Solve with method of Frobenius
     • Generating function
     • Integral formulas for Bessel functions with integer order
     • You may be asked to prove various identities and formulas derived in class.

Structure of the exam

There will be 5 to 7 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 3/7/07 (fjn).