= Solution
Take the closed surface with its outward orientation. For sufficiently smooth fields defined throughout the solid, the <divergence theorem> and the identity that the <divergence> of a <curl> is zero give
$$
I=\iiint_V\nabla\cdot(\nabla\Omega+\nabla\times\mathbf U)\,dV=\boxed{\iiint_V\Delta\Omega\,dV.}
$$
Thus the <curl> term carries no net outward <flux>, and $I$ is independent of $\mathbf U$. The hole in the <solid torus> does not invalidate the <divergence theorem>; singularities of $\mathbf U$ \i[inside the <integration> region] would invalidate the smoothness argument and must be excluded.
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