= Solution
The remaining terms give
$$
\boxed{(\nabla\times\mathbf f)\cdot\mathbf r=\frac1r\Delta_{\theta,\phi}F,\qquad\Delta_{\theta,\phi}=\frac1{\sin\theta}\partial_\theta(\sin\theta\,\partial_\theta)+\frac1{\sin^2\theta}\partial_\phi^2.}
$$
This is the <Laplace-Beltrami operator> on the unit <sphere>. Taking the <dot product> with the unit radial <vector> instead would give $r^{-2}\Delta_{\theta,\phi}F$; the extra factor $r$ here comes from the position <vector> in the question.
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