 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/b/solution

The radial component of the curl in spherical coordinates is

(\nabla \times f)_{r} = \frac{1}{r s i n θ} [\partial_{θ} (sin θ f_{ϕ}) - \partial_{ϕ} f_{θ}] .

It contains no

f_{r}

, so

R

cannot contribute. Substituting the tangential components gives

\partial_{θ} (sin θ f_{ϕ}) - \partial_{ϕ} f_{θ} = \frac{1}{r} [G_{ϕθ} - G_{θϕ} + \partial_{θ} (sin θ F_{θ}) + \frac{F _{ϕϕ}}{s i n θ}] .

The mixed partial derivatives of smooth

G

agree, so its terms cancel. Multiplication by

r

r = r e_{r}

, does not change this independence from

R, G

 New to topics? Read the docs here!