Divergence and curl of a radial vector field ($\nabla \cdot (f(r)x)=r{f}^{\prime }(r)+nf(r),\nabla \times (f(r)x)=0(n=3)$)

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 $f(r)=C{r}^{-n}$ on each positive radial interval. This formula does not assert regularity at the origin.