Divergence Theorem

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For a Differentiable function $\vec{F}:\mathbb{R}\to {\mathbb{R}}^n$, the divergence theorem states that, for a given closed surface $S$,

$$\underset{S}{\oint }\vec{F}\cdot \overrightarrow{dS}=\underset{V}{\iiint }\nabla \cdot \vec{F}dV$$

That is, the Flux of a vector field through a closed surface is equal to the Divergence of the vector field inside the surface’s volume.

Generalization for higher order spaces

Generalization in Geometric Algebra

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References