Math 642 Midterm Review — Spring 2019
The midterm will
consist of an in-class part, which will be given on April 1, and a
take-home part, which will be due Friday, April 5. 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, 6.5.2. Specific topics are listed below.
The in-class part, which will be worth 80 points (40%) 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; it will be
worth 120 points (60%).
Calculus of Variations
- Section 5.1
- Frechet and Gateaux derivatives, definitions.
- Be able to derive the Euler-Lagrange equations, using variational
calculations, for constrained and unconstrained problems that are
subject to various boundary conditions.
- Coordinate invariance of an extremal under change of
coordinates. Geodesics.
- Section 5.2
- Hamilton's principle; Lagrangians.
- Legendre transformations and Hamiltonians.
- Hamilton's equations; conserved quantities — angular
momentum, energy.
- 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
- Be able to use variation methods for finding eigenvalues and
eigenfunctions in a Sturm-Liouville problem, subject to various
boundary conditions.
- Minimum principle.
- Courant-Fischer minimax theorem; be able to sketch a proof.
Complex variables
- Section 6.2
- Cauchy's theorem, Cauchy's integral formulae.
- Taylor and Laurent series.
- Isolated singularities; classification: removable; pole of
order $m$; essential.
- Residues and the residue theorem.
- 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
- Jordan's lemma. Be able to apply it and be able to prove it.
- Contour integration, with and without cuts.
- Section 6.5
- Gamma function
- Definition and extension to $\mathbb C \setminus \{0,-1,-2,\cdots\}$
- Fundamental identity (Keener, Eqn. 6.25; class notes, 3/6 &
3/8). Be able to sketch a proof.
- 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.
Updated 3/29/19 (fjn).