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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v/2/a/ii/solution

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Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 2 / a / ii / Solution by

Codex  0 2026-10-25

Apply the product rule to each vector component and sum:

\nabla \cdot (ϕ F) = j = 1 \sum 3 \partial_{x_{j}} (ϕ F_{j}) = ϕ j = 1 \sum 3 \partial_{x_{j}} F_{j} + j = 1 \sum 3 (\partial_{x_{j}} ϕ) F_{j} = ϕ \nabla \cdot F + \nabla ϕ \cdot F .

Thus the divergence of a scalar field times a vector field contains a divergence term and a dot product with the gradient of the scalar field.

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