# Homework Help: Proof of Divergence Formula in Spherical Coordinates

1. Nov 18, 2007

### eliasds

Hello - I'm supposed to derive the divergence formula for spherical coordinates by carrying out the surface integrals of the surface of the volume in the figure (the figure is a piece of a sphere similar to a box but with curves). The radial coord is r. The polar angle is $$\varphi$$ and the azimuthal angle is $$\theta$$.

The divergence formula is easy enought to look up: DIV(F) = $$\nabla\bullet$$F =

$$\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}F_{r}$$+$$\frac{1}{rsin\varphi}\frac{\partial}{\partial \varphi}\left( sin\varphi F_{\varphi}\right)$$ + $$\frac{1}{rsin\varphi}$$$$\frac{\partial F_{\theta}}{\partial\theta}$$

And the volume of the little piece of a sphere is easy enough:
$$r^{2}sin\varphi \Delta r \Delta\varphi\Delta\theta$$

But when I try to set up the limits for each side as the volume goes to zero I never end up with the first and second $$sin\varphi$$ in the equation. Supposedly I'm supposed to multiply by a $$sin\theta$$ but I don't see why.

What I end up with is:
$$\frac{\partial}{\partial r}F_{r}$$+$$\frac{1}{r}\frac{\partial}{\partial \varphi}\left( F_{\varphi}\right)$$ + $$\frac{1}{rsin\varphi}$$$$\frac{\partial F_{\theta}}{\partial\theta}$$

Last edited: Nov 19, 2007
2. Nov 18, 2007

### EnumaElish

How is polar angle different from azimuth angle?

3. Nov 19, 2007

### eliasds

Phi is the angle in the page, and the theta is the angle out of the page.