 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-v-resit/question-2/a/ii/solution

For

r > 0

, the chain rule gives

\partial r / \partial x_{i} = x_{i} / r

. Applying the product rule to the components of the vector field yields

\nabla \cdot u = i = 1 \sum 3 \frac{\partial}{\partial x _{i}} (x_{i} r^{- α}) = i = 1 \sum 3 (r^{- α} - α x_{i}^{2} r^{- α - 2}) = 3 r^{- α} - α r^{- α - 2} i = 1 \sum 3 x_{i}^{2} = \frac{3 - α}{r ^{α}} .

The calculation is pointwise away from the origin; any extension of the vector field to the origin must be considered separately.

 New to topics? Read the docs here!