All submissions for homework 6 have been marked. I hope that between my comments and the homework solutions you are able to understand how to obtain the correct answer -- if you are still not sure, please come talk to either me or Dmitri in our respective office hours, or send me an e-mail.

A couple reminders from things that came up while I was marking the homework: (1) please remember that sin m pi = 0 for all integers m, and that cos m pi = (-1)^m for all integers m. (2) Integrals like INT_0^2pi sin m x sin n x dx, etc., are examples of inner products of elements of the orthogonal set { cos m x, sin m x } on [0, 2pi], and hence can be immediately seen to be zero unless m = n. (3) Final expansion coefficients in the series for a solution u of Laplace's equation cannot depend on spatial variables (x, y, z, r, theta, phi, rho, etc.). When doing a problem, please make sure that the final expansion coefficients you determine do not depend on any of these quantities as otherwise your answer is essentially guaranteed to be incorrect. (This does not apply to the intermediate expansion coefficients we use when solving three-dimensional problems, such as spherical problems without azimuthal symmetry or the second problem on homework 6.)

If any of the foregoing are mysterious, please talk to either myself or Dmitri before the test to get the matter clarified.

There will be no formula sheet for the final: if I feel that a particular formula is needed for a particular problem I will give it there. I have however put together a review sheet (available here) which contains a very dense overview of most of what we have done in the course so far. I have done my best to avoid errors but please let me know if you spot any!

I expect you to know pretty much everything on page 1 of the review sheet (in particular, you need to memorise the general forms for the solutions to Laplace's equation). As for page 2, I expect you to memorise the following: P_0, P_1, P_2, P_{1, 1}, the formula for P_{l, m} in terms of P_l, and the normalisation for J_m and P_l (not P_{l, m} for m nonzero); but the last three are less important. While I will not expect you to memorise the rest of the information on page 2, you should certainly be familiar with it as otherwise you will probably have difficulty applying it on the test.

I will put up more practice problems sometime tomorrow. I have posted Quiz 5 with solution here for use in preparing for the test.

Let me know if you have any questions!

Nathan