My tired, allergy impaired brain needs a bit of help. Last week I did the example problem for a loop of current, in the x-y plane, and finding the magnetic field on the z axis a height P above the loop. I did it in the traditional admixture of cartesian and polar coordinates to get:
dB = mu0 I [ az (cos(theta) xhat + sin(theta) yhat) + a^2 zhat] d(theta)/4*pi*R^3
If I integrate that from 0 to 2*pi the x and y terms drop out and I get the expected result.
Now, one of my students reworked the problem in cylindrical coordinates and (as far as I can tell) correctly gets:
dB = mu0 I [a z d(theta) rhat + a^2 d(theta) zhat] / 4*pi*R^3
If I integrate that from 0 to 2*pi I get the correct z term but the rhat term does not go to zero. I'm fairly certain it has to do with the limits of integration, but can't for the life of me figure out what they are.