I have corrected the following errors:

-- Final review sheet: the normalisation integral for the spherical Bessel functions had an extra factor of r inside the j_{ell + 1}; this has been removed. (The integral int_0^1 j_ell^2(kappa_{ell i} r) r^2 dr should be 1/2 j_{ell + 1}(kappa_{ell i}), not 1/2 j_{ell + 1} (kappa_{ell i} r).)

-- Final exam practice problems: problem 18 should probably have had a sin phi rather than a sin 2phi dependence since the theta integral doesn't seem to be workable otherwise.

Also, a brief comment about normalisations when using eigenfunction expansions to represent Green's functions. The general formula which we gave in class and in the lecture notes assumed that the eigenfunctions were already normalised; since the eigenfunctions which we use are not in general normalised, we have to divide by the same normalisation factor (e_I, e_I) which we have used when computing coefficients in orthogonal expansions throughout the course. This explains, for example, the factor 8 in the expression for the Green's function in the solutions to problem 1 of Homework 11.

I plan to post solutions to selected problems from the final exam review sheet sometime this evening. The main point in posting them so late is to make everyone try them on their own before looking at my solutions (as usual with practice problems). I anticipate being able to take questions until 11 PM tonight but cannot guarantee to answer any questions after that (including questions sent tomorrow morning) before the exam.

I will be in the MAC again at 4.00.

Let me know if you have any questions!

Nathan