= Solution
For a continuously differentiable <vector field> $\mathbf u$ on a bounded region $V\subset\mathbb R^3$ with piecewise smooth closed boundary $S=\partial V$, the <divergence theorem> states
$$
\boxed{\int_V\nabla\cdot\mathbf u\,dV=\int_S\mathbf u\cdot\hat{\mathbf n}\,dS=\int_S\mathbf u\cdot d\mathbf S.}
$$
Here $\nabla\cdot\mathbf u$ is its <divergence>, $dV$ is the volume element, $dS$ is the surface area element, $\hat{\mathbf n}$ is the outward <unit normal>, and $d\mathbf S=\hat{\mathbf n}\,dS$. Thus a <volume integral> of the <divergence> equals the outward <flux> through the boundary.
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