The divergence theorem

This theorem equates the integral of the divergence of a vector field $\underline{F}$ over a volume $v$ to the outgoing flux of the same vector field $\underline{F}$ across the closed regular surface $S$ surrounding $v$ (see Fig. (A.2)), according to the formula:

$$\boxed{ \int_V \nabla \cdot \underline{F} \, dV = \oint_S \hat{n} \cdot \underline{F} \, dS }\space ,$$ (2.15)

where $\hat{n}$ is the unit normal to S pointing outward from $V$.The divergence theorem is also known as Gauss' theorem.