## Review guide for the final exam

 Some Math 421exam axioms
 Exam information Math 421:01 Friday December 17 8 AM-11 AM SEC 216 Math 421:03 Tuesday December 21 8 AM-11 AM SEC 204 The final exam will be cumulative, but it will have somewhat more emphasis on material which has been covered since the last exam.
 Calculators, notes,formula sheets, etc.

Review sessions
Certainly all students are welcome at all review sessions. The sessions may work better if they are nominally devoted to specific topics. So I will offer to review certain topics and answer questions about these topics, and when that's done, offer to answer questions about other 421 material.

Last homework assignments I plan to spend the last class in 421:01 answering questions about the last homework assignment and the two dimensional PDE problems. Students from 421:03 who are free and want to participate are invited: Monday, December 13, 11:30-12:50, in SEC 216.

Wednesday, December 16 at 4-6 PM in Hill 425 Laplace transforms & linear algebra.

Thursday, December 17 at 4-6 PM in Hill 425 Fourier series, the wave/heat equations, & associated boundary value problems.

What's next?
The Goals below are copied from the syllabus. The remarks after Reality attempt to describe what sections in the text were actually covered and give links to suggested review material. What's here applies to both sections 1 and section 3 of Math 421 whose final exams should be similar.

Laplace transforms
Goals Students should know the definition of the Laplace transform. They should be able to compute simple Laplace transforms "by hand" and should be able to use a table of Laplace transforms to solve initial value problems for constant coefficient ODE's. They should be able to write "rough" data in terms of Dirac and Heaviside functions. They should be able to recognize and use simple results regarding Laplace transforms.

Reality We covered sections 4.1-6. Sources for review material:

Linear algebra
Goals Students should know the principal definitions of linear algebra and recognize and work with these ideas. Students should know how to compute the rank of a matrix, the determinant of a matrix, and be able to invert and diagonalize simple matrices "by hand".

Reality We covered sections 8.1-8.8, 8.10, and 8.12. Students should be able to state the principal definitions. Sources for review material:

• Last spring's course used a different text and covered slightly different topics. You can look at review problems and answers. See especially problem 14 and its answers. That problem asks for very brief definitions of common terms in linear algebra. I think students in this semester's class should also be able to answer that sort of question. A version of the second exam I gave and answers to that exam are available. You may look at these problems, too .
• The second exam and its answers which I gave this semester are available. Please note especially problem 3, which indicates my expectation that students should be able to apply the ideas of linear algebra to varied situations. The first 4 problems are relevant here.
Fourier series
Goals Students should be able to compute the Fourier coefficients of simple functions "by hand". They should know what orthogonality means for functions. Given a function's graph on [0,A] they should be able to graph the sum of the Fourier sine and cosine series on [-A,A]. They should have some qualitative understanding of what partial sums of Fourier series look like compared to the original function, including an idea of the Gibbs phenomenon.

Reality Certainly we discussed the material in sections 12.1-12.3. I will not ask any questions about 12.4 (complex Fourier series). The ideas of 3.9 and 12.5 are essential in the last portion of the course. The specific content needed from section 12.5 is brief. Sources for review material:

• The first seven review problems here cover material to be tested on this exam, and here are answers. Problems 1, 2, and 3 on this exam could be asked here.
• The second exam and its answers which I gave this semester are available. Problems 5, 6, and 7 are about Fourier series.
Boundary value problems for ODE's and PDE's
Goals Students should be able to solve simple boundary and initial value problems for the classical PDE's using separation of variables combined with the appropriate trigonometric series. They should also understand some of the qualitative aspects of solutions (smoothness for heat and possible shocks for wave), and simple asymptotics for some BVP's.

Reality Here the difference between what is declared in the syllabus and what I feel can realistically be covered is greater than in other parts of the semester. I think the exam should cover 13.1-5 and 13.8. I am writing this before the last week of the semester. I hope to cover in lecture some discussion of steady-state heat flows on a rectangular plate (Laplace's equation) and some discussion of vibrations on a rectangular plate. I haven't taught this before and I may need to revise my ambitions. Sources for review material:

• Again, the past spring semester has useful links. Problems 8 through 17 of this review set cover material to be tested on this exam, and here are answers. Problems 4 through 7 on this exam could be asked here.
• Here are some questions about two-dimensional material: heat, wave, and Laplace's equations, and double sine series. This material was not covered in the spring 2004 version of the course.

I try to write and grade exams carefully. As you prepare for and work on the final exam, I hope that you will:

1. Do my problems, not problems you invent.