Math 412 - Test II Review
General Information
Test II (Thursday, November 15) will have 5 to 7 questions, some with
multiple parts. Paper will be provided. Problems will
be similar to ones done for homework. 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.
Topics Covered
- Fourier Transforms - § 10.1-10.4, 10.6.1.
- Know how to find the Fourier transform of simple functions via
the definition.
- Know how to find the convolution of two functions and how to find
the inverse Fourier transform of a product via the Convolution
Theorem.
- Be able to use any of the properties from my web notes to find Fourier transforms
and inverse Fourier transforms. (You will be given a copy of these
notes.)
- Know how to derive simple Fourier transform properties similar to
ones we did in class or for homework.
- Know how to obtain the heat kernel and be able to solve the wave
equation via Fourier transform techniques.
- Wave equation & characteristics - § 12.1-12.3
- Be able to solve first order wave equations via the method of
characteristics.
- Be able to derive d'Alembert's solution to the wave equation. Be
able to use it to find solutions in special cases.
- Know what the domain of dependence and the domain of
influence are. Be able to find them in simple cases.
- Sturm-Liouville eigenvalue problems - § 5.1-5.8 (h > 0)
& 5.10.
- Be able to state the conditions for a problem to be a regular
Sturm-Liouville problem. Know the properties of eigenvalues and
eigenfunctions. (See § 5.3.2.)
- Be able to derive the Rayleigh quotient and to use it to show
reality and positivity of eigenvalues in specific problems.
- Know what a selfadjoint operator is, and what the Lagrange and
Green identities are. Be able to show that eigenfunctions
corresponding to distinct eigenvalues are orthogonal relative to a
weight function.
- Be able to estimate the lowest eigenvalue in a regular
Sturm-Liouville problem via the Rayleigh quotient.
- Be able to write out solutions to the heat and wave equations
when the eigenvalue problems involved are regular Sturm-Liouville
problems. (See § 5.7-5.8.)
- Know the approximation properties of eigenfunction expansions. Be
able to define mean-square error and to state Parseval's equation.