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 Incoming links: Weighted gradient circulation identity

Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 8 / b / Solution 2026-10-25

For a twice continuously differentiable scalar function

ψ

, equality of mixed partial derivatives gives

\nabla \times \nabla ψ = 0

. Substitute

F = \nabla ψ

into the preceding product rule:

\nabla \times (ϕ \nabla ψ) = \nabla ϕ \times \nabla ψ .

Stokes theorem on a spanning surface with consistent boundary orientation then gives

\oint_{\partial S} ϕ \nabla ψ \cdot d l = \iint_{S} (\nabla ϕ \times \nabla ψ) \cdot d 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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