Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 3A / i / Solution

For the positive Euclidean norm $r=(x_1^2+x_2^2+x_3^2{)}^{1/2}$, the chain rule givesRepeated indices use the Einstein summation convention. The origin is excluded because the Euclidean norm is not differentiable there. Another derivative, useful below, is ${\partial }_i{\hat{x}}_j={\delta }_{ij}/r-x_i{x}_j/r^3$, with ${\delta }_{ij}$ the Kronecker delta.