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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014/12/a/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2014 / 12 / a / Solution by

Codex  0 2026-10-25

The components of the curl of a gradient are differences of mixed second derivatives:

\nabla \times \nabla ϕ = (ϕ_{zy} - ϕ_{yz}, ϕ_{x z} - ϕ_{z x}, ϕ_{y x} - ϕ_{x y}) .

(1)

ϕ

has continuous second partial derivatives in a neighborhood, equality of mixed derivatives makes every component vanish. Thus

\nabla \times (\nabla ϕ) = 0.

(2)

The

C^{2}

condition is a sufficient regularity hypothesis; no simple-connectivity assumption is needed for this local identity. Topology matters for the converse assertion that every curl-free field has a global potential.

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