Gauss-Ostrogradsky Theorem/Intuitive Illustration

Proof
Consider a closed surface $S$ within a body of fluid which at an arbitrary point $P$ is in motion with velocity $\mathbf V$.

The total rate of flow of fluid passing through $S$ can be found in $2$ different ways:


 * $(1): \quad$ By calculating $\mathbf V \cdot \rd \mathbf S$, which is the product of a small element of $S$ and the component of velocity which is perpendicular to it, and then adding all these contributions


 * $(2): \quad$ By investigating the divergence of an element of volume, which is the excess of the sources of fluid over its sinks per unit volume and integrating $\operatorname {div} \mathbf V \rd v$ throughout the volume enclosed by $S$.

These $2$ results are physically equivalent, because the excess fluid leaving $S$ over that entering must be because of the fluid injected into $B$ by the aggregate of sources and sinks.

Also see

 * Green's Theorem