Let be a solid and be its oriented boundary.
Let be a vector field differentiable in .
Then calculus
Note the left hand side is equivalent to the flux through the surface of ,
the right hand side refers to Divergence.
Heuristically, if a region has no divergence, there is no nett in-flow or out-flow, and therefore the flux through the boundary is zero.
Corollaries
\begin{align*}
\iiint_{\Omega} (\vab{\nabla} \times \vab A) ,d\tau = -\oiint_{\partial\Omega} \vab A \times d\vab a
\end{align*}