Radial power-law flux through a spherical shell (${\int }_{\partial V}\mathbf{F}\cdot d\mathbf{A}=4\pi (R^{2-\gamma }-{ϵ}^{2-\gamma })$)

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.