= Solution
In cylindrical coordinates, the transverse velocity is $A\mathbf e_\theta/r$, so its contribution to the <circulation> is $A\,d\theta$. The vertical contribution is $U(z)\,dz$, the differential of an antiderivative of $U$, and therefore has zero integral around any closed curve. If $k$ is the oriented <winding number> of the curve's projection about the origin, then
$$
\boxed{\oint\mathbf u\cdot d\mathbf r=2\pi A k.}
$$
In particular, \b[a single positive winding gives $2\pi A$]; reversing orientation reverses the sign. The nonzero <circulation> is compatible with <irrotational flow> because the domain is not <simply connected>. For $A\ne0$, an enclosing curve cannot bound a surface contained entirely in the domain to which <Stokes theorem> would apply.
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