If I'm evaluating $\int^b_a r dz$, where $r$ is a variable in spherical coordinates and $z$ is a variable in rectangular coordinates, do I need to transform $dz$ in spherical coordinates? If so, how would I do that?
This question was spawned from integration of radiation fields for dipoles.
Thank you for your time.
Yes. In spherical coordinates: (Assuming that "theta" is the angle made with the positive x-axis, and "phi" is the angle made with the positive z-axis)
z = r*cos(phi)
Also:
x = r*sin(phi)*cos(theta), and:
y = r*sin(phi)*sin(theta)