Math 312-500--Review for Test II

General information. The test will be held on Thursday, April 15 at our normal class time and in our usual room. It will have 5 to 7 questions, some with multiple parts. The test will cover sections 1.9, 1.11C, 2.6-2.8, 2.10, 2.11, 3.1, 3.2, and 5.9 in Powers, along with the material covered in class: spherical harmonics, properties of the Fourier transform, the Sampling Theorem, and the Uncertainty Principle. Please bring an 8½x11 bluebook. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing either calculus or linear algebra. A table of integrals and a list of properties for Fourier transforms will be provided.

Separation of variables & eigenvalue problems. Be able to solve heat-flow and vibrating string problems via the method of separation of variables. (Sections 2.6, 2.10, 2.11, and 3.2) Be able to solve eigenvalue problems. Know what a regular Sturm-Liouville problem is. Be able to show orthogonality of the eigenfunctions, and be able to solve problems similar to the ones given for homework. (Sections 2.7 & 2.8)

Spherical harmonics. The spherical harmonics Y_l,m arise in problems having spherical symmetry. They are eigenfunctions for the Laplace-Beltrami operator on the sphere. Let t (for "theta") be the colatitude and p (for "phi") be the longitude. For m=0, Y_l,
0(t,p)=const*P_l(cos(t)), where P_l is a Legendre polynomial of degree l. In general,

Y_{l, m}(t,p)=const*sin^|m|(t) P^(|m|)_l(cos(t)) exp(i m p),

where m = -l ... l, l = 0,1,2,... . In spherical coordinates, the function r^lY_l,
m(t,p) is a solution to Laplace's equation. Be able to show orthogonality of the spherical harmonics, and be able to derive the differential equation for the Legendre polynomials. Also, be able to find the first few Legendre polynomials.

Fourier transforms. Be able to derive simpler properties of the Fourier transform, and to calculate Fourier transforms of functions either using these properties or via the definition. See your class notes for examples. Be able to state the Uncertainty Principle and to verify it for a given function. Be able to state the Sampling Theorem. Know the simple properties and definitions of the Fourier sine and cosine transforms, and how they are related to the Fourier transform itself. (Sections 1.9, 1.11C, 2.10 & 2.11 and class notes.)

Office hours. In addition to my usual 11-12 Wednesday office hours, I will have office hours from 1:30 to 4:30 pm on Wednesday, April 14.