Summary of the Divergence Theorem

The divergence theorem relates a surface integral across closed surface [latex]S[/latex] to a triple integral over the solid enclosed by [latex]S[/latex]. 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
[latex]\displaystyle\iiint_{E} \text{div }{\bf{F}}dV=\underset{S}{\displaystyle\iint}{\bf{F}}\cdot{d{\bf{S}}}[/latex]

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

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

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 [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}[/latex] depends only on the value of [latex]f[/latex] at the endpoints of [latex]C[/latex]: [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))[/latex]