divergence theorem

Trades a closed surface's total flux for a triple integral of divergence over the solid region it encloses.

The Divergence Theorem states $\oiint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E \operatorname{div}\mathbf{F}\,dV$ , needing $S$ to be a closed surface with no boundary at all (a sphere, a cube, any surface fully enclosing a solid $E$ ) — unlike Stokes' Theorem, which needs an open surface with a rim. Since $\operatorname{div}(\operatorname{curl}\mathbf{F})\equiv0$ identically, applying the theorem to $\operatorname{curl}\mathbf{F}$ over any closed surface gives $\oiint_S\operatorname{curl}\mathbf{F}\cdot d\mathbf{S} = 0$ .

Examples

For $\mathbf{F}(x,y,z) = (x,\,y,\,z)$ and $S$ the sphere of radius $2$ : $\oiint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E 3\,dV = 32\pi$ .
A water balloon with no leaks or taps inside has divergence zero throughout, so zero net flow crosses its own skin.