Solution
= Solution
Let $V\subset\mathbb R^3$ be a bounded region with a piecewise smooth closed boundary $S=\partial V$, and let $\mathbf F$ have continuous first partial derivatives on a neighborhood of its closure. The <divergence theorem> states
$$
\boxed{\int_V\nabla\cdot\mathbf F\,dV=\int_S\mathbf F\cdot\mathbf n\,dS,}
$$
where $\mathbf n$ is the outward unit <normal vector>. The right side is the outward <flux>; reversing the orientation reverses its sign.