 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016-resit/3/iii/solution

Away from the origin, the product rule and

\nabla r = r / r

give

\nabla \cdot (r^{- 3} r) = r^{- 3} (3) + r \cdot (- 3 r^{- 5} r) = 0 (r \neq = 0) .

The field is undefined at

r = 0

. It has outward flux

4 π

through any sphere centred there, so its global distributional divergence is

4 π δ^{(3)} (r)

, not the zero distribution. The pointwise result must not be extended through the singularity when applying the divergence theorem.

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