How to Transform Integrals from Cylindrical to Spherical Coordinates?

[ShayanJ]

Consider an integral of the type $\int_0^{a} \int_0^{\pi} g(\rho,\varphi,\theta) \rho d\varphi d\rho$. As you can see, the integral is w.r.t. cylindrical coordinates on a plane but the integrand is also a function of $\theta$ which is a spherical coordinate. So for evaluating it, there are two options: 1) Write $\theta$ in terms of cylindrical coordinates. 2) Transform the integral to spherical coordinates. The first option makes the integral an intractable mess. But the second option seems nice because the integrand(in spherical coordinates) contains a factor of the form $(1-2xt+t^2)^{-\frac 1 2}$ and so the integral can be done using Legendre polynomials and spherical harmonics. But I don't know how I should transform the integral from cylindrical to spherical coordinates. Can anyone help?
Thanks

 

[Svein]

I am not quite sure what you are asking for. If θ is independent of ρ and φ, what you get after the integration is a function of θ. Otherwise, you need to clarify, since neither cylindrical nor spherical coordinates make any sense in a plane (restricted to a plane, they are both polar coordinates).

 

[ShayanJ]

I am not quite sure what you are asking for. If θ is independent of ρ and φ, what you get after the integration is a function of θ. Otherwise, you need to clarify, since neither cylindrical nor spherical coordinates make any sense in a plane (restricted to a plane, they are both polar coordinates).

Of course $\theta$ depends on $\rho$ and $\varphi$. The only point is that, at some point, I had to use the formula $\hat r_1 \cdot \hat r_2=\cos\theta_1 \cos\theta_2+\sin\theta_1 \sin\theta_2 \cos(\varphi_1-\varphi_2)$ which is in spherical coordinates. But at other parts, I had to use cylindrical coordinates. Also the problem is 3 dimensional but the integration is done on a surface but I should write the distance between an arbitrary point on the surface and an arbitrary point of the space.

 

[Svein]

But I don't know how I should transform the integral from cylindrical to spherical coordinates. Can anyone help?

How about θ = arctan(z/ρ)?

 

[ShayanJ]

How about θ = arctan(z/ρ)?

Well, That's good but as I said, the integral is a big mess in cylindrical coordinates. But in spherical coordinates, I can do the integral using Legendre polynomials and spherical harmonics. So I want to know how can I transform the integral from cylindrical to spherical coordinates.
I'm just wondering because transforming integrals from Cartesian coordinates to spherical and cylindrical coordinates is really straightforward but it doesn't seem to be the case about transformation between cylindrical and spherical coordinates. Isn't there any reference about it?