 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/iii/solution

m

is constant, so

\nabla (m \cdot r) = m

. Also

\nabla (r^{- 3}) = - 3 r / r^{5}

, by the same chain rule. The product rule therefore yields

\nabla (\frac{m \cdot r}{r ^{3}}) = \frac{m}{r ^{3}} - \frac{3 ( m \cdot r ) r}{r ^{5}}, r > 0.

The three components follow by differentiating

(αx + β y + γ z) (x^{2} + y^{2} + z^{2})^{- 3/2}

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