Math 412 - Final Exam Review
General Information
The final exam (12:30-2:30 pm, Friday, December 7) will have 7 to 9
questions, some with multiple parts. Paper will be provided. Problems
will be similar to ones on homework and previous exams. 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.
Sections Covered
- Chapter 2: 2.1-2.3, 2.4.2, 2.5.2, 2.5.4
- Chapter 3: 3.1-3.6
- Chapter 10: 10.1-10.4, 10.6.1
- Chapter 12: 12.1-12.3
- Chapter 5: 5.1-5.8 (h > 0), 5.10
- Chapter 7: 7.1-7.4, 7.7, 7.8.3, 7.10.1-7.10.3
Topics Covered
- Separation of Variables
- Be able to use separation of variables to solve heat flow
problems, wave equations, Laplace's equation, and similar problems. A
summary of the technique is given in §2.3.8. Here are the
pertinent sections.
- Heat flow problems. §§2.3, 2.4.2, 5.2, 5.4, 5.8, 7.1,
7.2.2, 7.3 (exercises).
- Wave equations. §§5.7, 5.8, 7.3, 7.7.
- Laplace's equation. §§2.5.2, 2.5.4. Also, know the mean-value
theorem, maximum principle, and definition of a well-posed problem.
- Helmholtz equation (eigenvalue problems). §§7.4, 7.7,
7.10.2.
- Fourier Series
- Be able to find the Fourier series, Fourier cosine series, and
Fourier sine series for functions. Know the convergence theorems for
these series. §§3.1, 3.2, 3.3, 3.6.
- Be able to briefly describe the Gibbs' phenomenon. §3.3.1.
- Be able to use symmetry properties to simplify finding
coefficients, and to compute coefficents via differentiation and
integration of a known series. (Basics only!) §§3.4, 3.5.
- Fourier Transforms
- Conventions for the Fourier transform and inverse Fourier
transform are given in web
notes, and so is a list of properties. (You will be given the same
list as you had on the last exam).
- Know how to find the Fourier transform or inverse Fourier
transform of simple functions. §§10.3, 10.4, and exercises.
- 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 Fourier transform techniques to solve
problems similar to obtaining the heat kernel or deriving the solution
the wave equation in an infinite "string". §§10.4.3, 10.6.1.
- Wave equation & characteristics
- Be able to solve first order wave equations via the method of
characteristics. §12.2.
- Be able to apply d'Alembert's solution to the wave equation to
find solutions in special cases. §12.3.2.
- Sturm-Liouville eigenvalue problems
- Be able to state the conditions for a problem to be a regular
Sturm-Liouville problem. Know the properties of eigenvalues and
eigenfunctions. § 5.3.2.
- Be able to write out solutions to the heat and wave equations
when the eigenvalue problems involved are regular Sturm-Liouville
problems. §§5.4, 5.7, 5.8.
- Know what a selfadjoint operator is. Be able to show that
eigenfunctions corresponding to distinct eigenvalues are orthogonal
relative to a weight function, even for singular Sturm-Liouville
problems. §5.5
- Be able to derive the Rayleigh quotient and to use it to show
reality and positivity of eigenvalues in specific problems. Know how to
estimate the lowest eigenvalue in a regular Sturm-Liouville problem
via the Rayleigh quotient. §5.6.
- Be able to define mean-square error and to use Parseval's
equation. §5.10.
- Problems in two and three space dimensions
- Be able to separate out the time in heat and wave equations to
arrive at an eigenvalue problem involving a Helmholtz
equation. §§7.2, 7.4.
- Know how to solve the wave equation in a rectangle and to make
nodal diagrams. §§7.3, 7.4.
- Be able to obtain eigenfunctions and eigenfrequencies in
vibrations in a circular drumhead, including the relationship to
Bessel functions. §7.7.
- Know the difference between Bessel functions of the first and
second kind (Jm and Ym). Be able to solve
Bessel's equation via series. See §§7.7.4-7.7.7 and class
notes for 27 November.
- Know what the spherical harmonics are. See class notes 29
November and §7.10.3.