= Solution
For a twice continuously differentiable scalar function $\psi$, equality of mixed <partial derivatives> gives $\nabla\times\nabla\psi=0$. Substitute $\mathbf F=\nabla\psi$ into the preceding product rule:
$$
\nabla\times(\phi\nabla\psi)=\nabla\phi\times\nabla\psi.
$$
<Stokes theorem> on a spanning surface with consistent boundary orientation then gives
$$
\boxed{\oint_{\partial S}\phi\nabla\psi\cdot d\mathbf l=\iint_S(\nabla\phi\times\nabla\psi)\cdot d\mathbf S.}
$$
This is a <weighted gradient circulation identity>. All fields must be defined and sufficiently smooth on a neighborhood of the surface; a scalar prefactor can create nonzero circulation even when the unweighted field is a gradient.
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