## Math 442 Final Exam, Spring 2008

Wednesday 7 May, 1:30-4:30pm, worth 45%.
You may use books, notes, and written materials, but no electronic devices.

Office hours before the exam: Monday 5 May noon-1pm, Tuesday 6 May noon-1pm.

Study session Tuesday 6 May, 5-6pm, in 345 Altgeld (usual classroom)

Material
The entire course. Up to half the exam will be on material covered since Tests 1 and 2 (that is, on Chapter 6 and later). All material covered in class and in homework is examinable.

How to study

• First make summary notes of the important ideas and methods from each section.
• Review the Test 1 and Test 2 information sheets for advice on those parts of the course. Following are pointers on the later material, covered since those tests.
• Ask yourself questions like "What are the most important examples of harmonic functions, in 2 and 3 dimensions?"
• "What does the maximum principle say for solutions of Laplace's equation? for solutions of the diffusion equation? and what are some applications of the maximum principle, for each equation?"
• "How do we solve Laplace's equation in a rectangle in two dimensions? a box in 3 dimensions? a disk in 2 dimensions? a ball in 3 dimensions?" (For rectangles and boxes, you may suppose the BC is either Dirichlet or Neumann, on each side. For disks you may suppose the BC is either Dirichlet on the whole circle or is Neumann on the whole circle, and similarly for balls.)
• "What does the Mean Value Property say? and what is an interesting application of it?"
• (Aside: we have solved Poisson's equation in 3 dimensional space, but not in 2 dimensional space and not in bounded domains in 2 or 3 dimensions.)
• Be familiar with separation of variables in: xy-coordinates, in polar coordinates and in spherical coordinates, for: Laplace's equation (which does not involve t) and also for the diffusion and wave equations (which do involve t).
• Work through relevant examples and exercises in the text.
• Where relevant, express the conclusions of each section as algorithms or checklists: step 1, step 2, and so on, so that you have a plan of action for each type of problem.
• Classify all homework problems: what type of PDE, BC, IC is involved? Then on the exam, start each problem similarly, by classifying.