= Solution
For a bounded <volume> $V$ with outward-oriented piecewise smooth boundary $S$, the <divergence theorem> states
$$
\boxed{\iint_SG\cdot n\,dS=\iiint_V\nabla\cdot G\,dV.}
$$
Here $G$ is a continuously differentiable <vector field> on a neighborhood of $V$, $n$ the outward unit <normal vector>, and $dV$ the <volume element>. The left side is its outward <flux>.
For a twice continuously differentiable field $F$, equality of mixed <partial derivatives> gives
$$
\nabla\cdot(\nabla\times F)
=(F_{z,yx}-F_{y,zx})+(F_{x,zy}-F_{z,xy})+(F_{y,xz}-F_{x,yz})=0.
$$
Apply the <divergence theorem> to $G=\nabla\times F$ to obtain
$$
\boxed{\iint_S(\nabla\times F)\cdot d\boldsymbol S=0.}
$$
The regularity hypotheses justify differentiating and integrating the <curl>; a field singularity cannot simply be ignored in a volume argument.
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