 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013-resit/8/b/solution

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 .

\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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