Gauss's law

Remarks

The proof reduces to one point charge by superposition and Fubini's theorem. For a charge $q$ at $y$, the field $q (x - y) / (4 \pi \varepsilon_0 \abs{x - y}^3)$ has divergence zero away from $y$. By the divergence theorem, its flux through $\partial \Omega$ equals its flux through a small sphere around $y$ if $y \in \Omega$, which is $q / \varepsilon_0$, and it is $0$ if $y \notin \overline{\Omega}$. For point charges, the same holds when no charge lies on $\partial \Omega$: the flux is the sum of the charges inside $\Omega$ divided by $\varepsilon_0$. Example: a ball of radius $R$ with uniform total charge $Q$ has, at distance $r > R$ from its centre, a radial field of size $Q / (4 \pi \varepsilon_0 r^2)$, by symmetry and the integral form.

Going deeper

For a student

Gauss's law is the integral form of the equation $\nabla \cdot E = \rho / \varepsilon_0$, which is the first of Maxwell's equations. It is equivalent to Coulomb's law for static fields, once one also knows that the field is curl-free. With symmetry, it gives the field of spheres, cylinders and planes without any integration. Combined with $E = -\nabla \varphi$, it yields Poisson's equation for the electric potential.