Principles and Techniques of Applied Mathematics
MATH 373, Final Exam

Saturday, May 5, 2:30 - 5:30 pm (Swords 328)

The final exam is cumulative, spanning all the material we covered in the first 5 chapters of the course text. It is highly recommended that you go over homework problems, your class notes and the two midterm exams. Some good sample problems from the text covering Ch. 5 are included below. You should also refer to previous exam review sheets for other study questions.

Note: You will be allowed one "cheat sheet," a single 8.5 x 11 piece of paper, front and back, full of whatever formulas, graphs, etc. you wish.

Exam Review: We will review for the final exam on Thursday, May 3rd from 2:00 - 3:30 pm in Swords 328. Please come prepared with specific questions.

The following concepts, definitions, equations, formulae, theorems and corollaries are important material for the exam:

1. General PDE terminology: operators, linear operators, homogeneous and inhomogeneous equations, classifying second-order linear PDE's (hyperbolic, elliptic, parabolic), steady-state or static solutions (no time dependence), harmonic functions, Principle of Superposition

2. Important PDE's: Heat equation, Wave equation, Laplace's equation, (know each in different spatial dimensions), the Laplacian (in standard coordinates, cylindrical, polar, spherical), physical interpretation of a given PDE, connections to music via the wave equation (vibrating string versus the vibrating drum)

3. Initial and Boundary Conditions: physical interpretations of initial conditions (position, velocity, etc.), describing given conditions mathematically, boundary conditions (Dirichlet, Neumann, Robin, periodic), physical interpretations of boundary conditions, flux, normal derivative, Newton's law of cooling, homogeneous vs. inhomogeneous conditions

4. Solving PDE's: Finding steady-state or static solutions in one and two spatial dimensions (eg. on a line segment, on a rectangle or on a disk), general solution to the wave equation, d'Alembert's formula for the wave equation on the entire real line (no boundary conditions), separation of variables technique for solving boundary value problems (in one and two spatial dimensions), change of variables techniques (eg. using the steady-state solution to reduce the PDE to a homogeneous case), solving PDE's with different boundary conditions (Dirichlet, Neumann, mixed, periodic) as infinite series, solving Sturm-Liouville problems (finding eigenvalues and eigenfunctions)

5. Analysis: piecewise continuous functions, piecewise smooth functions, left and right-hand limits, left and right-sided derivatives, definition of continuity, definition of the derivative, infinite series (definition of convergence, nth-term test, harmonic series, p-series, series of functions), the Fundamental Theorem of Calculus (FTC), even and odd functions, chain rule (single and multivariable versions), solving 2nd-order constant coefficient ODE's (characteristic polynomial, Euler's formula, etc.)

6. Fourier Series: definitions of, finding coefficients of (for cosine, sine and full Fourier series on the standard intervals (0,Pi) or (-Pi,Pi) or on the non-standard intervals (0,c), (-c,c)), linearity of Fourier coefficients, how to go from a Fourier series on a standard interval to one on a non-standard interval, double Fourier series

7. Convergence Theorems: even periodic extensions, odd periodic extensions, general periodic extensions, Fourier coefficients go to zero, Bessel's inequalities, the Dirichlet kernel, the Fourier convergence theorem and its corollaries, convergence on non-standard intervals, Gibbs phenomenon

Some Practice Problems from Ch. 5:

Chapter 5
pp. 108 - 109 :   3
pp. 113 - 114 :   5
pp. 122 - 124 :   2, 3, 6   (Note: Assume Neumann BC's for x=0 and x=c in problem #6.)
pp. 128 - 129 :   1
pp. 133 - 134 :   2, 3, 5
pp. 143 :   2
pp. 146 - 147 :   2