 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-2/16/a/ii/solution

For

r = x^{2} + y^{2} + z^{2} > 0

, the chain rule gives

\partial_{x} (r^{- 1}) = - x / r^{3}

, with analogous formulas for the other components. Hence

\nabla (1/ r) = - \frac{r}{r ^{3}} .

On the right the numerator denotes the position vector, while its denominator is the cube of its magnitude.

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