= Solution
The radial component of the <curl> in <spherical coordinates> is
$$
(\nabla\times\mathbf f)_r=\frac1{r\sin\theta}\left[\partial_\theta(\sin\theta f_\phi)-\partial_\phi f_\theta\right].
$$
It contains no $f_r$, so $R$ cannot contribute. Substituting the tangential components gives
$$
\partial_\theta(\sin\theta f_\phi)-\partial_\phi f_\theta=\frac1r\left[G_{\phi\theta}-G_{\theta\phi}+\partial_\theta(\sin\theta F_\theta)+\frac{F_{\phi\phi}}{\sin\theta}\right].
$$
The mixed <partial derivatives> of smooth $G$ agree, so its terms cancel. Multiplication by $r$, to take the <dot product> with the position <vector> $\mathbf r=r\mathbf e_r$, does not change this independence from $R,G$.
Back to article page
