= Solution
To obtain the <volume integral of a curl>, test its components with an arbitrary constant vector $c$. By part (a), $\nabla\cdot(G\times c)=c\cdot(\nabla\times G)$. Applying the <divergence theorem> to the closed boundary, with outward <unit normal> $n$, gives
$$
c\cdot\iiint_V\nabla\times G\,dV
=\iint_S(G\times c)\cdot n\,dS
=c\cdot\iint_S n\times G\,dS.
$$
The last step is cyclic symmetry of the <scalar triple product>. Since this holds for every $c$, all vector components agree:
$$
\boxed{\iiint_V\nabla\times G\,dV=\iint_Sn\times G\,dS.}
$$
The same regularity conditions as the <divergence theorem> apply: for example a continuously differentiable field on a neighbourhood of a bounded volume with piecewise smooth boundary.
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