 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-v-resit/question-3/b/solution

Apply the product rule inside the Cartesian expression for the curl:

\nabla \times (ζ v) = i \sum e_{i} \times \partial_{i} (ζ v) = i \sum e_{i} \times ((\partial_{i} ζ) v + ζ \partial_{i} v) = (i \sum (\partial_{i} ζ) e_{i}) \times v + ζ i \sum e_{i} \times \partial_{i} v .

\nabla \times (ζ v) = ζ \nabla \times v + \nabla ζ \times v .

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