Test 2 Review Math 412-200 (Summer II, 2011)
General Information
Test 2 (Tuesday, August 2) will have 5 to 7 questions, some with
multiple parts. The problems will be similar to homework problems,
examples done in class and examples worked out in the text. I will
have extra office hours on Monday afternoon, 1-4 pm. (No Tuesday
morning office hours.)
-
Calculators. You may use scientific calculators to do numerical
calculations logs, exponentials, and so on. You
may not use any calculator that has the capability of doing
algebra or calculus, or of storing course material.
-
Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
-
Coverage. The material covered is from the following sections in
the text: 4.4, 12.3, 5.1-5.8, 5.10.
Topics Covered
- Vibrating string & wave Equation -- §4.4 and
§12.3
- Be able to use separation of variables to solve the wave equation
for the vibrating string, subject to various boundary conditions and
initial conditions.
- Know what normal modes are (p. 144-145) and be able to work
problems similar to those in the homework.
- Energy in a vibrating string. See problems 4.4.9 and
4.4.13(a). Also, see Parseval's theorem (§5.10 and notes, July
27.)
- Know d'Alembert's form of the solution to the wave equation, u(x,t)
=F(x − ct) + G(x + ct). Be able to use it to find solutions in
special cases.
- Sturm-Liouville eigenvalue problems - § 5.1-5.8 and
5.10.
- Be able to state the conditions for a problem to be a regular
Sturm-Liouville (SL) problem. Know the properties of eigenvalues and
eigenfunctions. (See § 5.3.2.)
- Be able to solve heat-flow and vibrating-string problems
involving general SL problems. (See §5.4 and the notes for
July 25.)
- Know what a self-adjoint operator is. Be able to derive the
Lagrange and Green identities. Be able to show that eigenfunctions
corresponding to distinct eigenvalues are orthogonal, relative to the
weight function σ(x). (HW 5 and §5.5)
- The Rayleigh quotient. Be able to derive it and to use it to show
that in certain problems the eigenvalues are nonnegative. Be able
to use it to estimate eigenvalues in the two ways described below.
- The minimum principle. Estimate the lowest eigenvalue by plugging
in a suitable function that satisfies the boundary conditions in the
problem. (§5.6 and notes for July 26.)
- Bound the lowest eigenvalue by eigenvalues from known
problems. §5.7.
- Heat flow problems with boundary conditions derived from Newton's
law of cooling (Robin b.c.). §5.8, h > 0 only.
- Know the mean-square error and approximation properties of
expansions in orthogonal eigenfunctions. Also, be able to state
Bessel's inequality and Parseval's equation. Be able to use
Parseval's equation to show that the energy in a vibrating string is
the sum of the energy in each of its modes (July 27). Be able to use
Parseval's theorem to sum series. §5.10 and HW 5.
Updated: August 1, 2011 (fjn)