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.

Let $V=\{ϵ<r<R\}$ with $ϵ>0$ and $\mathbf{F}=\hat{\mathbf{r}}/r^{\gamma }$. Direct differentiation gives $\nabla \cdot \mathbf{F}=(2-\gamma )r^{-\gamma -1}$. For $\gamma \ne 2$, integration against $4\pi r^{2}dr$ gives the displayed expression. Independently, the two boundary spheres contribute $4\pi R^{2-\gamma }$ and $-4\pi {ϵ}^{2-\gamma }$, since the outward normal of the shell at its inner boundary points toward the origin. For $\gamma =2$, the divergence is zero in the shell and the surface contributions cancel exactly. The excluded origin must not be silently added to this regular-domain calculation.