Gauss-Ostrogradsky Theorem

Theorem
Suppose $$U \ $$ is a subset of $$\R^3 \ $$ which is compact and has a piecewise smooth boundary. If $$F:\R^3 \to \R^3 \ $$ is a smooth vector function defined on a neighborhood of $$U \ $$, then we have


 * $$\iiint\limits_U\left(\nabla\cdot\mathbf{F}\right)dV=\iint\limits_{\part U} \mathbf{F} \cdot \mathbf{n}\ dS \ $$

where $$\mathbf{n} \ $$ is the normal to $$\partial U \ $$.