 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/a/solution

Since

\hat{r} = e_{r}

e_{r} \times \nabla F = - \frac{F _{ϕ}}{r s i n θ} e_{θ} + \frac{F _{θ}}{r} e_{ϕ} .

(e_{r} \times \nabla G) \times e_{r} = \nabla G - e_{r} G_{r} = (G_{θ} / r) e_{θ} + [G_{ϕ} / (r sin θ)] e_{ϕ}

. Adding the radial part yields

f = R e_{r} + \frac{1}{r} (G_{θ} - \frac{F _{ϕ}}{s i n θ}) e_{θ} + \frac{1}{r} (\frac{G _{ϕ}}{s i n θ} + F_{θ}) e_{ϕ} .

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