= Solution
Take any constant <vector> $\mathbf a$. Its <curl> is zero, so part (a) and the <divergence theorem> give
$$
\mathbf a\cdot\int_V\nabla\times\mathbf F\,dV
=-\int_V\nabla\cdot(\mathbf a\times\mathbf F)\,dV
=-\int_S(\mathbf a\times\mathbf F)\cdot\mathbf n\,dS,
$$
where $\mathbf n$ is the outward unit <normal vector>. The <scalar triple product> identity rewrites the last integrand as $\mathbf a\cdot(\mathbf F\times\mathbf n)$. Since $\mathbf a$ is arbitrary, equality of these <scalar products> implies the <vector> identity
$$
\boxed{\int_V\nabla\times\mathbf F\,dV
=-\int_S\mathbf F\times\mathbf n\,dS
=\int_S\mathbf n\times\mathbf F\,dS}.
$$
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