A sufficient criterion for smoothness at a cylindrical axis

Statement

Let $a(s,z),b(s,z),c(s,z)$ be smooth on an open neighborhood of the relevant points with $s\ge0$ . An axisymmetric field on $r>0$ with

u_r=r\,a(r^2,z),\qquad u_\theta=r\,b(r^2,z),\qquad u_z=c(r^2,z)

extends smoothly through the axis.

Cartesian verification

Putting $s=x^2+y^2$ , its Cartesian components are

u_x=x\,a(s,z)-y\,b(s,z),\qquad u_y=y\,a(s,z)+x\,b(s,z),\qquad u_z=c(s,z).

These are compositions and products of smooth functions. This proves the stated sufficient criterion directly, including parameter-dependent versions when $a,b,c$ are jointly smooth in those parameters.

Why radial smoothness alone is insufficient

The field $e_r=(x/r,y/r,0)$ has constant cylindrical radial component but no continuous extension at zero. Even a scalar $f(r)=r$ , smooth on the radial half-line in its one-sided sense, becomes $\sqrt{x^2+y^2}$ , which is not differentiable at the axis.