= Solution
{c}
Choose $\mathbf F=\nabla\psi$ in the preceding <product rule> and apply the <divergence theorem>. Since $\nabla\cdot\nabla\psi=\Delta\psi$, this gives the first of <Green's identities>:
$$
\boxed{\iiint_V\bigl(\phi\Delta\psi+\nabla\phi\cdot\nabla\psi\bigr),dV
=\iint_{\partial V}\phi\,\partial_n\psi\,dS.}
$$
Here $\partial_n\psi=\mathbf n\cdot\nabla\psi$ is the outward <normal derivative>. Interchange $\phi$ and $\psi$, then subtract. The <gradient> <dot products> cancel, yielding the second of <Green's identities>:
$$
\boxed{\iiint_V(\phi\Delta\psi-\psi\Delta\phi)\,dV
=\iint_{\partial V}(\phi\partial_n\psi-\psi\partial_n\phi)\,dS.}
$$
For example, $\phi,\psi\in C^2(\overline V)$ with the needed extensions suffice for these applications of the <divergence theorem>.
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