Poisson's equation for gravitational field

[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?
Thanks in advance

 

[Orodruin]

Did you try using the divergence theorem?

 

[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

 

[Orodruin]

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

 

[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.

 

[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??

 

[Orodruin]

Yes, so what is your question?

 

[Sphinx]

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