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Deriving Gaussian Gravity

  1. Jul 11, 2008 #1
    1. The problem statement, all variables and given/known data
    How would I derive Gauss' law for gravity from Newton's law?


    2. Relevant equations

    [tex]\mathbf{g}(\mathbf{r}) &=& -G\frac{m_1}{{\vert \mathbf{r}\vert^2}}\hat{\mathbf{r}}[/tex]

    to

    [tex]\nabla\cdot\mathbf{g} = -4\pi G \rho[/tex]


    3. 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.

    [tex]\mathbf{g}(\mathbf{r}) = -G\int_V \frac{\rho(\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3} dV(\mathbf{s})[/tex]

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

    V
     
  2. jcsd
  3. Jul 11, 2008 #2
    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').
     
  4. Jul 11, 2008 #3

    nicksauce

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