## Math 442 Test 1, Spring 2008

Monday 18 February, in class, worth 15%.
You may use books, notes, and written materials, but no electronic devices.
Practice Test 1 and Solutions.

Study Session
Sunday 17 February, 4-5pm, in 345 Altgeld Hall.

Material
Sections 1.1-1.5 and 2.1-2.5, and all homework on that material. Use the Reading Guides, your lecture notes, the in-class worksheets and the homework as a guide to the emphasis you should place on the various topics. Some test questions will build on the homework. Some questions will be new.

How to study

• First make summary notes of the important ideas and methods from each section. (This effort forces you to mentally organize the material!)
• Where relevant, express the conclusions of the 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.
• If you just work problems without first making summary notes, you will probably be wasting your time, because you won't have a mental framework into which to fit the examples you are working.
• Re-work all homework problems.
• Work through relevant examples and exercises in the text.

A few basic suggestions

• Sections 1.3 and 1.4: I will not ask you to derive the diffusion or wave equation from first principles. But I might ask you to write down an equation that describes a reasonably familiar physical situation, such as "diffusion with a source and with one end of the test tube closed off and the other end immersed in a reservoir of concentration g(t) at time t."
• Sections 2.2 and 2.3: know the definitions of "energy" for the wave equation and for the diffusion equation, respectively, and know how to prove energy is conserved (wave equation) or dissipated (diffusion equation).
• Section 2.4: Exercises 2.4.16 to 2.4.19 are all suitable for test preparation.
• Exercise 2.4.19(b) is not clearly stated. I suggest you interpret it straightforwardly by showing that the integral of S2(x,y,t) dxdy equals 1, for each t, and that as t approaches zero, the area under the graph of S2 all concentrates at the origin. Since also S2 is positive and satisfies the two dimensional diffusion equation, we can interpret it as being the concentration at time t due to a unit of mass released at the origin at time zero.
A more sophisticated interpretation would be to prove an analogue in two dimensions of formula (6) on page 47.