 Solution

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

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

Away from the origin, differentiate

r^{2} = x_{j} x_{j}

2 r \partial_{i} r = 2 x_{i} .

Hence

\partial_{i} r = x_{i} / r, \nabla r = x / r .

This gradient is the radial unit vector. Division by

r

is legitimate on the punctured domain; the distance function is not differentiable at the origin.

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