= 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 $R$,
$$
\Gamma=\int_0^{2\pi}\frac\kappa{2\pi R}R\,d\theta=\kappa,\qquad M=0,
$$
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>. Consequently
$$
\boxed{\Gamma=\kappa,\qquad M=0.}
$$
Equivalently, $\Gamma=(\kappa/(2\pi))\oint d\theta$ 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>.
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