 Solution

Define the line vortex strength $\kappa$ by $\mathbf{u}=\kappa {\mathbf{e}}_{\theta }/(2\pi r)$, with $\kappa >0$ meaning counterclockwise motion. On an enclosing circle of radius $R$,because the velocity is tangent to the circle. For any simple positively oriented enclosing curve, the same results follow by Stokes theorem and the divergence theorem on the region between that curve and a small inner circle. Both curl and divergence vanish throughout that intervening annulus. ConsequentlyEquivalently, $\Gamma =(\kappa /(2\pi ))\oint d\theta$ and the total angle change is $2\pi$. If strength is instead denoted by $K$ in $u_{\theta }=K/r$, the answer is $\Gamma =2\pi K$. An enclosing disk through the singular origin cannot be used to infer zero circulation from zero local vorticity.