Test 2 Review Math 401-501/502 (Spring 2013)
General Information
Test 2 (Tuesday, 4/23/13) will have 4 to 6 questions, some with
multiple parts. The problems will be similar
to homework
problems, examples done in class and examples worked out in the
text. My office hours for the next few days are as follows: Thursday
(4/18/13), 2:30-3:30; Friday (4/19), 1:30-3:30; Monday (4/22/13),
10-1:30.
-
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.
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Formulas: You will be given a table of Fourier transforms
(Table A.2, plus a couple more), and a few integrals.
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Coverage: The test will be over material covered from after
Test 1 on through Thursday, 4/18/13. This includes chapter 2,
sections 5.1, 5.2, 5.3 (Laplace equation in a circular disk, p. 105),
5.5 (Vibrating circular membrane, p. 117), 8.1.
Topics Covered
- Fourier series
- Be able to find Fourier series, Fourier sine series, and
Fourier sine series for various functions. Be able to use symmetry
to calculate Fourier coefficients.
- Fourier's convergence theorem. Be able to define piecewise
continuous and piecewise smooth functions. Know how to extend
functions 2L periodic extensions; 2L even periodic
extensions; 2L odd periodic extensions. Be able to graph the
functions the series converge to, and know the three types of
convergence that we discussed in class (notes, 3/28) and for
homework
(see
assignments 6 and 7).
- Pointwise convergence jumps; Gibbs' phenomenon.
- Uniform only corners, no jumps; visual.
- Mean convergence energy of residual goes to 0; only need
finite energy to begin with.
- Parseval's identity
assignments
6 and 7.
- Separation of variables
- Be able to separate variables to solve the heat equation in a bar
of finite length, the wave equation for a finite string, and the
potential equation in a
disk. See assignment
8.
- Be able to solve eigenvalues problems associated with separation
of variables. Be able to obtain the Rayleigh quotient for these
eigenvalue problems. Here is an example of how to do that. Consider
this eigenvalue problem: $\ X''+ \lambda X = 0$, $X'(0)=0$,
$X(1)=0$. Do the following steps.
- Multiply by the equation by $X$: $XX''+ \lambda X^2=0$.
- Integrate: $\int_0^1 XX''dx + \lambda \int_0^1 X^2dx =0$
- Integrate the first integral by parts and use the boundary
conditions:
\[
\int_0^1 XX''dx = XX'\bigg|_0^1 - \int_0^1 X'^2 dx
=\underbrace{X(1)}_{0} X'(1) -
X(0)\underbrace{X'(0)}_{0} - \int_0^1 X'^2 dx = - \int_0^1 X'^2 dx
\]
- Substitute this for the integral on the right in step (b) and
solve for $\lambda$. The result is the Rayleigh quotient for the
problem,
\[
\lambda = \frac{\int_0^1 X'^2 dx}{\int_0^1 X^2 dx}.
\]
- Be able top obtain the normal modes, in terms of Bessel
functions, for the vibrating drumhead. Follow my notes from 4/16/13
and 4/18/13. Be able to derive the Rayleigh quotient in the this
case. I will write up notes recapping how to do this. (In the
text, section 5.5, p. 117, is good, but it covers more material than
I want to.)
- Fourier transforms
- Know how to find the Fourier transform or inverse Fourier
transform of simple functions. These will be similar to the ones
in assignment
9.
- Be able to use Fourier transform techniques to solve problems
similar to those for a heat equation in an infinite bar and an
infinite vibrating string. Again,
see assignment
9. Be able to derive the heat kernel. Know what the error
function is and be able to use it to solve heat flow problems, given
the solution in terms of the heat kernel.
Updated: April 17, 2013 (fjn)