divergence theorem

The divergence theorem is a generalization of the one-dimensional fundamental theorem of calculus:

\begin{align*} \int_{V}\vec{\nabla} \cdot \vec{f}(\vec{r})d\tau = \oint_{S}\vec{f}(\vec{r}) \cdot d\vec{a} \end{align*}

it is a statement about conservation in physical systems: the outflow of force from the inside of a closed boundary has to be equal to the amount of force at the boundaries.