 Weighted gradient circulation identity

For sufficiently smooth scalar fields near an oriented surface, the product rule for the curl gives $\nabla \times (\varphi \nabla \psi )=\varphi \nabla \times \nabla \psi +\nabla \varphi \times \nabla \psi =\nabla \varphi \times \nabla \psi$. Stokes theorem then proves the displayed identity with consistent boundary orientation. A scalar multiple of a gradient need not itself be a gradient; the cross product of the two gradients detects its local circulation.