Gauss's Divergence Theorem

Suppose $V$ is a subset of ${\mathbb{R}}^n$ which is compact and has a piece wide smooth boundary $S$ (also indicated with $\delta V=S$). If $\mathbf{F}$ is a continuously differentiable vector field defined on a neighborhood of $V$, then

$${\iiint }_V(\nabla \cdot \mathbf{F})\mathrm{d}V={\iint }_S(\mathbf{F}\cdot \hat{\mathbf{n}})\mathrm{d}S$$

The left side is a volume integral over the volume $V$, and the right side is the surface integral over the boundary of the volume $V$. The closed, measurable set $\delta V$ is oriented by outward-pointing normals