= Solution
Apply the <divergence theorem> to $\mathbf F=\psi\nabla\phi$, for functions smooth enough on the closure of the bounded piecewise smooth volume. The <product rule> gives
$$
\nabla\cdot(\psi\nabla\phi)=\nabla\psi\cdot\nabla\phi+\psi\nabla^2\phi,
\qquad (\psi\nabla\phi)\cdot\mathbf n=\psi\frac{\partial\phi}{\partial n}.
$$
Thus <Green's first theorem>, also called <Green's first identity>, is
$$
\boxed{\int_V(\psi\nabla^2\phi+\nabla\psi\cdot\nabla\phi)\,dV
=\int_{\partial V}\psi\frac{\partial\phi}{\partial n}\,dS.}
$$
The normal is outward. This identifies the boundary term and its sign directly from the <divergence theorem>.
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