# Poisson's equation for gravitational field

1. Jul 30, 2015

### Sphinx

Hello everyone!!
I have a question concerning the proof of Poisson's equation for a gravitational field.
My question is how can i prove that Gauss's Law is independent of the choice of the closed surface surrounding the point mass?

2. Jul 30, 2015

### Orodruin

Staff Emeritus
Did you try using the divergence theorem?

3. Aug 1, 2015

### Sphinx

Yes, but this is the last step of the proof , my difficulty resides in proving that the integral doesn't depend on the closed surface

4. Aug 1, 2015

### Orodruin

Staff Emeritus
It would easier to help you if you actually showed what you did.

5. Aug 1, 2015

### ChrisVer

You can either use the divergence theorem (the thing is that if you can prove it for a sphere you can prove it for any closed surface), or become hardcore (for no reason) and work with infinitesimal masses.

Last edited: Aug 1, 2015
6. Aug 2, 2015

### Sphinx

∫ƒ.ds over a closed surface =∫div(f) dΩ over the corresponding volume =∫div(f) dΩ over a sphere enclosing the volume =∫ƒ.ds over the surface of the sphere
is that correct??

7. Aug 2, 2015

### Orodruin

Staff Emeritus
Yes, so what is your question?

8. Aug 2, 2015

### Sphinx

I didn't see how to use the div theo, but know it's clear.
Thank u very much Orodruin & ChrisVer