= Solution
In three dimensions the <curl> can be written with the <Levi-Civita symbol>:
$$
(\nabla\times\mathbf F)_i
=\epsilon_{ijk}\partial_j(f(r)x_k)
=\epsilon_{ijk}\left(\frac{f'(r)}r x_jx_k+f(r)\delta_{jk}\right)=0.
$$
Both terms contract an antisymmetric array with a symmetric one, and therefore vanish. \b[Every differentiable radial field of this form has zero <curl> away from the origin.] This is the three-dimensional part of the <divergence and curl of a radial vector field>.
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