Deriving Gauss' Law for Gravity from Newton's Law

[Varnick]

Homework Statement

How would I derive Gauss' law for gravity from Newton's law?

Homework Equations

$$\mathbf{g}(\mathbf{r}) &=& -G\frac{m_1}{{\vert \mathbf{r}\vert^2}}\hat{\mathbf{r}}$$

to

$$\nabla\cdot\mathbf{g} = -4\pi G \rho$$

The Attempt at a Solution

I have no reference material outside the wide and bountiful internet, and wikipedia gives this equation as the first step of the derivation, which is where I'm really stuck.

$$\mathbf{g}(\mathbf{r}) = -G\int_V \frac{\rho(\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3} dV(\mathbf{s})$$

I'm just not sure how on Earth I'd get to this equation, any help appreciated.

V

 

[badphysicist]

I think that you'll need to use the divergence theorem (look it up on wikipedia). On the wikipedia webpage you can actually look at this problem under the applications part ('Gauss's law for gravity').

 

[nicksauce]

It may help at some point if you know that
$$\nabla\cdot\frac{\vec{r}}{r^2} = 4\pi\delta^3({\vec{r}})$$