= Weighted gradient circulation identity
{title2=$\oint_{\partial S}\phi\nabla\psi\cdot d\mathbf l=\iint_S(\nabla\phi\times\nabla\psi)\cdot d\mathbf S$}
For sufficiently smooth scalar fields near an oriented <surface>, the product rule for the <curl> gives $\nabla\times(\phi\nabla\psi)=\phi\nabla\times\nabla\psi+\nabla\phi\times\nabla\psi=\nabla\phi\times\nabla\psi$. <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.
Back to article page
