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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/ia/paper-3/3a/b/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 3 / 3A / b / ii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

In three dimensions the curl can be written with the Levi-Civita symbol:

(\nabla \times F)_{i} = ϵ_{ijk} \partial_{j} (f (r) x_{k}) = ϵ_{ijk} (\frac{f ^{'} ( r )}{r} x_{j} x_{k} + f (r) δ_{jk}) = 0.

(1)

Both terms contract an antisymmetric array with a symmetric one, and therefore vanish. Every differentiable radial field of this form has zero curl away from the origin. This is the three-dimensional part of the divergence and curl of a radial vector field.

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