= Divergence and curl of a radial vector field
{title2=$\nabla\cdot(f(r)x)=rf'(r)+nf(r),\quad\nabla\times(f(r)x)=0\ (n=3)$}
For $r=|x|>0$, the <gradient> satisfies $\partial_i r=x_i/r$. The <product rule> and <Einstein notation> then give the displayed <divergence> formula in dimension $n$. In dimension three, its derivative <matrix> is symmetric, so the <curl> vanishes. Zero <divergence> forces $f(r)=Cr^{-n}$ on each positive radial interval. This formula does not assert regularity at the origin.
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