Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V resit / Question 3 / b

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- Solution 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 .

The first sum is the gradient of the scalar field and the second is the curl of the vector field. Thus the curl product rule for a scalar field is

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

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