= Solution
{c}
Apply the <divergence theorem> to $\phi\mathbf c$, using <ii>. Because the <vector> $\mathbf c$ is constant, it can be taken outside both <integrals>:
$$
\mathbf c\cdot\iiint_V\nabla\phi\,dV
=\iiint_V\nabla\cdot(\phi\mathbf c)\,dV
=\iint_{\partial V}\phi\mathbf c\cdot\mathbf n\,dS
=\mathbf c\cdot\iint_{\partial V}\phi\mathbf n\,dS.
$$
This holds for every $\mathbf c$, in particular each of the three Cartesian <basis vectors>. Equality of the three components proves the <gradient volume-to-boundary identity>
$$
\boxed{\iiint_V\nabla\phi\,dV=\iint_{\partial V}\phi\mathbf n\,dS.}
$$
It suffices that $\phi$ be a <continuously differentiable function> on a neighbourhood of the <closure> of $V$.
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