Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 3 / 3A / a

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- Solution a

Solution

Away from the origin, differentiate

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

using Einstein notation:

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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