Summary of the Divergence Theorem

The divergence theorem relates a surface integral across closed surface $S$ to a triple integral over the solid enclosed by $S$ . The divergence theorem is a higher dimensional version of the flux form of Green’s theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus.
The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa.
The divergence theorem can be used to derive Gauss’ law, a fundamental law in electrostatics.

Key Equations

Divergence theorem
$\displaystyle\iiint_{E} \text{div }{\bf{F}}dV=\underset{S}{\displaystyle\iint}{\bf{F}}\cdot{d{\bf{S}}}$

divergence theorem: a theorem used to transform a difficult flux integral into an easier triple integral and vice versa

Gauss’ law: if $S$ is a piecewise, smooth closed surface in a vacuum and $Q$ is the total stationary charge inside of $S$ , then the flux of electrostatic field $\bf{E}$ across $S$ is $Q|{\varepsilon}_{0}$

inverse-square law: the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge

The Fundamental Theorem for Line Integrals: the value of the line integral $\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}$ depends only on the value of $f$ at the endpoints of $C$ : $\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))$