 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/9/b/solution

\nabla \cdot r = 3

, and the result in (a) give, for

r > 0

\nabla \cdot \frac{r}{r} = \frac{1}{r} \nabla \cdot r + r \cdot \nabla \frac{1}{r} = \frac{3}{r} - \frac{r \cdot r}{r ^{3}} = \frac{2}{r} .

This is a pointwise identity away from the origin, where the radial unit vector is undefined.

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