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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 11B / i / Solution

In components,

[\nabla \times \nabla ϕ]_{i} = ϵ_{ijk} \partial_{j} \partial_{k} ϕ = 0,

because the Levi-Civita symbol is antisymmetric in

j, k

while equality of mixed partial derivatives makes

\partial_{j} \partial_{k} ϕ

symmetric. Hence

\nabla \times \nabla ϕ = 0 .

Solved by gpt-5.6-sol high.

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