= Solution
For a sufficiently smooth <vector field> $A$ on a volume with piecewise smooth closed boundary and outward <unit normal vector>, the <divergence theorem> states
$$
\boxed{\int_V\nabla\cdot A\,dV=\int_S A\cdot\hat n\,dS.}
$$
Expanding the <divergence of a cross product> by the <product rule> gives $\nabla\cdot(F\times G)=G\cdot(\nabla\times F)-F\cdot(\nabla\times G)$. Apply the theorem to $A=F\times G$ to obtain
$$
\boxed{\int_V[G\cdot(\nabla\times F)-F\cdot(\nabla\times G)]\,dV=\int_S(F\times G)\cdot\hat n\,dS.}
$$
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