Question: Let
$G$ be a vector field in $B=\mathbb{R}^n-p-q$,
$\operatorname{div} G=0$ in $B,$
$N$ is the unit normal to $\partial M$ pointing outward.
As $M$ ranges over all compact $n$-manifolds in $\mathbb{R}^n$ for which $p,q\notin \partial M$, how many values does the integral $\int_{\partial M}\langle\, G,N \rangle \, dV$ have?
Solution/help needed:
By the divergence theorem,
$$\int_{\partial M}\langle\, G,N \rangle \, dV=\int_M \operatorname{div}G,$$
Therefore, clearly, if $p,q\notin M$, the integral is zero. However, I don't know how to deal with the case when $p, q,$ or both are inside the manifold.
Edit:
Suppose $p\in M$. Take a ball $B_\epsilon^n(p)$ of radius $\epsilon$ around $p$. Then, we break the integral:
$$\int_{M}\operatorname{div}G=\int_{M-B_\epsilon^{n}(p)}\operatorname{div}G+\int_{B_\epsilon^n(p)}\operatorname{div}G =0\,+\,\int_{S_\epsilon^{(n-1)}(p)}\langle\, G,N \rangle \, dV.$$ N is the normal unit vector field on the sphere of radius $\epsilon$ centered at $p$ (which is the boundary of $B_\epsilon^n(p)$). How should I continue from here? (I'm guessing that it should be a constant times the volume of the sphere, which the constant would be equivalent to the charge in the answer below.)
I would guess 4 (from physics); put unit charges +2 and -1 on p and q respectively. (and consider the vector field as the electric field that consequently forms). Then, you can consider the 4 cases: having none, having only p inside, having only q inside, and having both inside.
The following derivation of Gauss's law from Coulomb's law can probably help you with coming up with constructing example where 4 is tight.
https://physics.stackexchange.com/questions/38404/how-is-gauss-law-integral-form-arrived-at-from-coulombs-law-and-how-is-the