Inverse Square Law and various space dimensions

[Lapidus]

I am interested in the derivation of the inverse square law in various dimensions via Green's functions. I think the trick is to imagine a sphere and then to integrate over it. Does anyone know a book or notes where this is explained?

I found this below from here, but could not really understand what's going on:

Is this some theorem that connects the two integrals (Green's theorem possibly?) in (1.7)?

thanks

 

[Orodruin]

Is this some theorem that connects the two integrals (Green's theorem possibly?) in (1.7)?

The divergence theorem. (Gauss's law)