= Solution
Since $\hat{\mathbf r}=\mathbf e_r$, the <gradient> formula gives
$$
\mathbf e_r\times\nabla F=-\frac{F_\phi}{r\sin\theta}\mathbf e_\theta+\frac{F_\theta}{r}\mathbf e_\phi.
$$
The <vector triple product identity> gives $(\mathbf e_r\times\nabla G)\times\mathbf e_r=\nabla G-\mathbf e_rG_r=(G_\theta/r)\mathbf e_\theta+[G_\phi/(r\sin\theta)]\mathbf e_\phi$. Adding the radial part yields
$$
\mathbf f=R\mathbf e_r+\frac1r\left(G_\theta-\frac{F_\phi}{\sin\theta}\right)\mathbf e_\theta+\frac1r\left(\frac{G_\phi}{\sin\theta}+F_\theta\right)\mathbf e_\phi.
$$
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