= Solution
The <divergence of a cross product> and $\nabla\times\mathbf a=0$ give $\nabla\cdot(\mathbf a\times\mathbf A)=-\mathbf a\cdot(\nabla\times\mathbf A)$. On the boundary the <scalar triple product> gives $(\mathbf a\times\mathbf A)\cdot\hat{\mathbf n}=-\mathbf a\cdot(\hat{\mathbf n}\times\mathbf A)$. Thus the <divergence theorem> gives
$$
-\mathbf a\cdot\int_V\nabla\times\mathbf A\,dV=-\mathbf a\cdot\int_S\hat{\mathbf n}\times\mathbf A\,dS.
$$
Because $\mathbf a$ is arbitrary, the <curl> identity follows:
$$
\boxed{\int_V\nabla\times\mathbf A\,dV=\int_S\hat{\mathbf n}\times\mathbf A\,dS.}
$$
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