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.