= Solution
Let $V\subset\mathbb R^3$ be a bounded region with piecewise smooth closed boundary $\partial V$, let $\mathbf n$ be its outward <unit normal vector>, and let $\mathbf A$ be a continuously differentiable <vector field> on a neighbourhood of its closure. The <divergence theorem> states
$$
\boxed{\int_V\nabla\cdot\mathbf A\,dV=\int_{\partial V}\mathbf A\cdot\mathbf n\,dS.}
$$
Here $dV$ is the positive volume measure, $dS$ is the surface area measure, $\nabla\cdot\mathbf A=\partial_xA_x+\partial_yA_y+\partial_zA_z$, and the right side is the total outward <flux>. All boundary components must be included, with their normals pointing away from $V$.
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