Test 1 Review Math 412-200 (Summer II, 2011)
General Information
Test 1 (Tuesday, July 19) 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, and on Tuesday
morning, 8:30-9:30 am.
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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.
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Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
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Coverage The material covered is from the following sections in
the text: 1.1-1.5, 2.1-2.3, 2.4.2, 2.5.2-2.5.4, 3.1-3.5.
Topics Covered
- Heat Equation.
- Be able to derive the equation and know what the boundary
conditions mean, physically. (1.2-1.3)
- Be able to find steady-state (equilibrium) solutions. (1.4)
- Know the 2D Laplace operator in polar coordinates. (1.5)
- Solving Heat Flow and Laplace Equation Problems
- Separation of Variables. There are three steps to the
procedure. (See § 2.3)
- Working only with the partial differential equation itself,
assume that a solution is a product, e.g., φ(x)G(t) in the heat
equation for a bar. Find the differential equations φ and G
satisfy.
- Ignore any nonhomogeneous conditions in the problem. Use the
homogenous conditions to find the boundary conditions φ must
satisfy; set up the eigenvalue problem.
- Solve the eigenvalue problem and solve for the corresponding
G's. Write out the general solution to the problem.
- Eigenvalue problems. Be able to solve any of the eigenvalue
problems we have covered, including the one in § 2.4.1.
- Be able to derive/use orthogonality relations to solve heat flow
or Laplace equation problems. (§§2.4.2, 2.5.3)
- Mean value theorem and max/min principle. Be able to prove the
mean value theorem and the max/min principle for Laplace's
equation. Know Hadamard's three conditions for a well-posed
problem. (§2.5.4)
- Fourier series
- Be able to find Fourier series, Fourier cosine series, and
Fourier sine series for various functions. Know the connections
among these series. (§§3.1-3.3)
- Fourier's convergence theorem. Be able to define piecewise
continuous and piecewise smooth functions. Know how extend functions
2L periodic extensions; 2L even periodic extensions; 2L odd
periodic extensions. Be able to graph the functions the series
converge to. Know what the Gibbs' phenomenon is. Know the trhee
types of convergence we are using (pointwise, uniform,
mean). (§§3.1-3.3)
- Integration and differentiation of Fourier series. (§§3.4,
3.5)
Updated: July 14, 2011 (fjn)