 Solution

Write $\mathbf{F}=\mathbf{r}{r}^{-\gamma -1}$. For $r>0$, direct Cartesian differentiation gives the divergence and curl of a radial vector field:The volume element in spherical coordinates therefore gives, for $\gamma \ne 2$,For the direct surface integral, the outer sphere has outward normal $+\mathbf{n}$ and contributes $4\pi R^{2-\gamma }$. The inner boundary's outward normal relative to the shell is $-\mathbf{n}$, so it contributes $-4\pi {ϵ}^{2-\gamma }$. HenceFor $\gamma =2$, the field has zero ordinary divergence everywhere in this shell. The separate surface contributions are $4\pi$ and $-4\pi$, so both full integrals are zero. This radial power-law flux through a spherical shell was obtained by direct differentiation and integration; the excluded origin is never part of the volume.