 Solution

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

c = (c_{1}, c_{2}, c_{3})

. The product rule and constancy of its components give

\nabla \cdot (ϕ c) = \partial_{x} (c_{1} ϕ) + \partial_{y} (c_{2} ϕ) + \partial_{z} (c_{3} ϕ) = c_{1} ϕ_{x} + c_{2} ϕ_{y} + c_{3} ϕ_{z} .

The last expression is the dot product with the gradient, so

\nabla \cdot (ϕ c) = c \cdot \nabla ϕ .

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