= Cylindrically radial divergence
{title2=$\nabla\cdot[f(\rho)\mathbf e_\rho]=\rho^{-1}(\rho f)\prime$}
For a <vector field> directed radially from a fixed axis and depending only on distance $\rho$ from that axis, <cylindrical coordinates> give $\nabla\cdot[f(\rho)\mathbf e_\rho]=f\prime+f/\rho$. Zero <divergence> gives $f=A/\rho$ on any positive radial interval. Unless $A=0$, this field is singular on the axis, so applications of the <divergence theorem> must use a domain avoiding it or account separately for that singularity.
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