 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1/1/ii/c/solution

The remaining terms give

(\nabla \times f) \cdot r = \frac{1}{r} Δ_{θ, ϕ} F, Δ_{θ, ϕ} = \frac{1}{sin θ} \partial_{θ} (sin θ \partial_{θ}) + \frac{1}{sin ^{2} θ} \partial_{ϕ}^{2} .

This is the Laplace-Beltrami operator on the unit sphere. Taking the dot product with the unit radial vector instead would give

r^{- 2} Δ_{θ, ϕ} F

; the extra factor

r

here comes from the position vector in the question.

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