= Solution
{c}
For each of the three constant Cartesian <unit vectors> $\mathbf e_j$, apply the <divergence theorem> to the <vector field> $\phi\mathbf e_j$. Its <divergence> is $\partial_j\phi$, so
$$
\iiint_R\partial_j\phi\,dV
=\iint_{\partial R}\phi n_j\,dS
\qquad(j=1,2,3).
$$
These are precisely the three components of
$$
\boxed{\iiint_R\nabla\phi\,dV
=\iint_{\partial R}\phi\mathbf n\,dS
=\iint_{\partial R}\phi\,d\mathbf S}.
$$
Thus the <volume integral> of the <gradient> is the boundary <surface integral> weighted by the <vector surface element> $d\mathbf S=\mathbf n\,dS$. The regularity needed here is that $\phi$ be continuously differentiable on a neighbourhood of the <closure> of $R$.
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