Principles and Techniques of Applied Mathematics
MATH 373, Exam #1

Wednesday, Feb. 23, 1:00 - 2:30 pm

The first exam covers the first two chapters of the course text, Chapter 1 (Where PDE's Come From) and Chapter 2 (Waves and Diffusions), except for Section 1.6. It is highly recommended that you go over homework problems (solutions are on reserve in the black "Roberts" binder in the Math/Sci. Library) and your class notes.

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 exam on Monday, Feb. 21st from 6:00 - 7:30 pm in Swords 359. Please come prepared with specific questions.

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

  1. General PDE terminology: operators, linear operators, homogeneous and inhomogeneous linear equations, the order of a PDE, well-posed problems (existence, uniqueness and stability)
  2. Initial and Boundary Conditions: physical interpretations of initial conditions (position, velocity, etc.), describing conditions mathematically, boundary conditions (Dirichlet, Neumann, Robin), physical interpretations of boundary conditions
  3. First-order Linear Equations: solving (geometric method or the method of characteristics, coordinate method), solving to satisfy an initial condition, solving inhomogeneous linear equations (use u_h + u_p), characteristic curves
  4. The Wave Equation: factoring an operator, general solution, D'Alembert's formula, physical interpretation of solutions, principle of causality, conservation of energy
  5. The Diffusion Equation: the Maximum (and minimum) Principle, properties of solutions (eg. invariance, well-posed for t > 0 , etc.), physical interpretation of solutions, general solution, the source function S(x,t), the error function Erf(x)
  6. Important Mathematical Concepts: Fundamental Theorem of Calculus, differentiating an integral, chain rule (multivariable version), the Divergence Theorem, the Laplacian operator, the gradient operator
  7. Important Physical Concepts: F = ma, physical interpretations of the partial derivative, force vectors, integrating to calculate mass, equilibrium solutions

Some Practice Problems:

Chapter 1
Section 1.1 :   1, 11
Section 1.2 :   2, 7, 11
Section 1.3 :   6, 10
Section 1.5 :   2

Chapter 2
Section 2.1 :   2, 7, 9
Section 2.2 :   2, 5
Section 2.3 :   2, 4a, 4b, 6
Section 2.4 :   2, 3, 5, 7