Math 312-500--Final Exam Review

General information. The test will be held on Friday, May 7 from 3 to 5 pm in our usual room. It will have 6 to 9 questions, some with multiple parts. The test will cover this material: Tolstov, chapters 1 and 2 -- except for sections 1.4, 2.8, and 2.9 --; Powers, sections 1.9, 1.11C, chapter 2, sections 3.1-3.3, 3.6, 4.4, 4.5, 5.7, and 5.9; and, the material from the lectures on the spherical harmonics, properties of the Fourier transform, the Sampling Theorem, and the Uncertainty Principle. Please bring two 8½x11 bluebooks. 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. The material covered after Test II will count for approximately 20% of the final exam. We review this material below. For the material covered prior to Test II, see the review sheets for Tests I & II.

The wave equation and d'Alembert's method. Be able to solve vibrating string problems in infinite or semi-infinite cases via d'Alembert's method. Be able to derive d'Alembert's solution to the wave equation in one space dimension. Also, be able to use Fourier transforms, Fourier sine transforms, and Fourier cosine transforms to solve the wave equation in the infinite and semi-infinite cases. (Sections 3.3 & 3.6)

The potential equation. We solved the potential equation for the disk and for the sphere, using polar and spherical coordinates. Both equations were subject to Dirichlet boundary conditions (u is given on the boundary). Be able to solve similar problems in the disk and in the sphere. (See section 4.6 and the class notes.)

The vibrating drumhead. We found the normal modes for solutions to the wave equation in a disk, subject to Dirichlet boundary conditions (u=0 on the edge of the disk). We also described nodal diagrams for this problem.