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Take $0<r_1<r_2$, so the shell excludes the possible singularity at the origin. The divergence and curl of a radial vector field formula in spherical coordinates givesWith volume element $dV=r^2\sin \theta drd\theta d\varphi$, direct integration givesFor the independent boundary calculation, the outer sphere has outward unit normal vector $+\hat{\mathbf{r}}$ and the inner sphere has outward unit normal vector $-\hat{\mathbf{r}}$. Their fluxes areTheir sum equals the direct volume integral, verifying the divergence theorem. The restriction $\beta \ne 2$ permits the displayed power-antiderivative step. At $\beta =2$, treated separately, both the shell volume integral and net boundary flux are zero; the excluded origin must still not be filled in.

On the outward-oriented sphere of radius $R$, $\mathbf{n}=\mathbf{x}/R$, so $\mathbf{x}\cdot d\mathbf{S}=RdS$. HenceAway from the origin, the divergence and curl of a radial vector field with $f(r)=r^{-3}$ giveIf the enclosed volume excludes the origin and its boundary avoids it, this field is smooth throughout the volume. The divergence theorem then gives $I=0$. The nonzero sphere flux is compatible with zero divergence away from the origin because the enclosed singularity prevents applying that smooth-volume theorem directly to the full ball.

In three dimensions the curl can be written with the Levi-Civita symbol:Both terms contract an antisymmetric array with a symmetric one, and therefore vanish. Every differentiable radial field of this form has zero curl away from the origin. This is the three-dimensional part of the divergence and curl of a radial vector field.

Apply the product rule and the preceding gradient formula to the vector field $F_i=f(r)x_i$. Summing over the repeated index givesHere ${\delta }_{ii}=n$ for the Kronecker delta. Thus the divergence and curl of a radial vector field give, in any dimension,

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.