= Solution
Away from the axis, <cylindrical coordinates> put the field in the <line vortex> form
$$
\mathbf v=\frac{\kappa}{2\pi\rho}\mathbf e_\phi.
$$
The only potentially nonzero component of its <curl> is
$$
(\nabla\times\mathbf v)_z=\frac1\rho\frac{\partial}{\partial\rho}(\rho v_\phi)=\frac1\rho\frac{\partial}{\partial\rho}\frac\kappa{2\pi}=0.
$$
The other components vanish because there is no axial dependence and no radial or axial velocity. Thus \b[the <curl> is zero for $\rho>0$]; the field is singular on the axis, so a classical <curl> there is not defined.
For a spanning surface avoiding that axis, the smoothness hypotheses of <Stokes theorem> hold and give $\oint_C\mathbf v\cdot d\mathbf r=0$. If the enclosed planar region contains the axis, the singularity prevents applying that theorem directly. Excise a small disk about the origin. The resulting annulus has zero <curl>, so its outer <circulation> equals the counterclockwise circulation on the small circle:
$$
\boxed{\oint_C\mathbf v\cdot d\mathbf r=\int_0^{2\pi}\frac{\kappa}{2\pi\rho}\rho\,d\phi=\kappa.}
$$
This sign assumes positive, counterclockwise orientation. Clockwise orientation gives $-\kappa$; more generally the value is $\kappa$ times the <winding number> about the origin. A curve through the origin is excluded because its integral is singular. In the <distribution> interpretation the missing <vorticity> is concentrated on the axis as $\kappa\delta(x)\delta(y)\mathbf e_z$, explaining the nonzero circulation despite zero classical <curl> away from it.
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