I have finally finished reading through your solutions, and tried to provide some comments. I apologise for the delay this week and will be back on schedule next week.

Some general comments (which applied to multiple papers, so I might not have written it on your paper even if it applied):

-- It is not correct to say "m = 1" in problem 1, or "m = 2" in problem 3 -- the only thing we can say is that the coefficients a_mn and b_mn vanish unless m = 1 or m = 2, respectively. m is just a dummy variable which only exists underneath the summation sign. It is also not correct to write out different solutions for different values of m -- the given problems have only one solution.

-- On that note, please remember that final expansion coefficients cannot depend on spatial variables, and also that when we take an inner product it is always the integral over the entire interval with which we are working. (Sometimes we can split the resulting integral up into multiple different integrals, but the result is still just one inner product -- it is incorrect to try to split up the inner product itself (before writing it out as an integral).) I have talked about this multiple times before but it still seems to be an issue; if you aren't sure why this is so, or don't understand what the point is, please write or come talk to me or Dmitri to get it cleared up. (If you make this kind of mistake on the final, you are likely to lose a lot of marks, so please do get in touch with us if you don't understand it!)

-- In class and/or section today we will go over the alternate form for the general solution to Laplace's equation in cylindrical coordinates (the same thing applies for spherical coordinates) which I discussed on Thursday. In particular, it turns out that this form is necessary to solve problem 1 in terms of a single series.

-- In the assignment, only problem 3 required the use of multiple series in the setup. Problem 1 can be solved using two different series but that wasn't necessary. (This relates to the previous point; also check the solutions.)

Finally, I presume that you are comparing your solutions to the solutions I post on the webpage, whether or not I ask you to do so in my comments.

Office hours, tutorials, quizzes, and class will be held today as usual. In class today we will talk about finding the eigenvalues and eigenfunctions for the Laplacian on the unit cube in R^3. I can't seem to find any readings in the textbook which cover this exactly.

See you all this afternoon!

Nathan