 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ia/paper-3/3a/i/solution

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

r = (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})^{1/2}

, the chain rule gives

\partial_{i} r = \frac{1}{2} (x_{j} x_{j})^{- 1/2} (2 x_{i}) = \frac{x _{i}}{r}, \nabla r = x .

Repeated indices use the Einstein summation convention. The origin is excluded because the Euclidean norm is not differentiable there. Another derivative, useful below, is

\partial_{i} x_{j} = δ_{ij} / r - x_{i} x_{j} / r^{3}

, with

δ_{ij}

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