Electric Flux out of Closed Surface surrounding Point Charge

Theorem
Let $q$ be a point charge.

Let $S$ be a closed surface surrounding $q$.

The total electric flux through $S$ is given by:
 * $F = \dfrac q {\varepsilon_0}$

Lemma
From this lemma, we can transform the surface integral for the flux through the entire surface $S$ into a double integral between angles: