I"d suggest switching to polar coordinates at some point: compute. Bonus [about 5 points]: find the jacobian j = j(x, y, z) for the change of variables from spherical to rectangular coordinates. Note: this is the reverse of the usual change. Z z z f d d d = z z z f j dxdydz. Remarks and answers: the average was about 65 based on the top 20 scores, which is a fairly normal result, but down a bit from exam 1 and exam 2. The two highest scores were 97 and 78. The results were high on problems 1 and 2, but lower on the polar coordinate problems, 3, 4, and 6, and on the proof. I have written your semester exam average into the upper right, as usual, with an estimate of your current grade (not yet including hw or optional projects).
