= 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>:
$$
\partial_i(x_i r^{-\gamma-1})=3r^{-\gamma-1}-(\gamma+1)r^{-\gamma-3}x_ix_i=(2-\gamma)r^{-\gamma-1}.
$$
The volume element in <spherical coordinates> therefore gives, for $\gamma\ne2$,
$$
\boxed{\int_V\nabla\cdot\mathbf F\,dV=4\pi(2-\gamma)\int_\epsilon^R r^{1-\gamma}dr=4\pi(R^{2-\gamma}-\epsilon^{2-\gamma}).}
$$
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 $-4\pi\epsilon^{2-\gamma}$. Hence
$$
\boxed{\int_S\mathbf F\cdot d\mathbf A=4\pi(R^{2-\gamma}-\epsilon^{2-\gamma}).}
$$
For $\gamma=2$, the field has zero ordinary divergence everywhere in this shell. The separate surface contributions are $4\pi$ and $-4\pi$, so \b[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.
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