= Solution
An enclosing loop has no spanning disk wholly inside the fluid, so initial zero <vorticity> does not force its <circulation> to vanish. <Kelvin's circulation theorem> says only that the initial <circulation> is retained:
$$
\boxed{\Gamma(C(t))=\Gamma(C(0)),\quad\text{possibly nonzero}}.
$$
For example $\mathbf u=\Gamma_0\mathbf e_\theta/(2\pi r)$ has zero <vorticity> outside the cylinder but <circulation> $\Gamma_0$ around a once-enclosing positively oriented loop. Any two such loops have equal <circulation>, since their difference bounds an annulus in the fluid and <Stokes theorem> gives zero over it. More generally a loop with winding number $m$ has <circulation> $m\Gamma_0$. Thus <irrotational flow> need not have a single-valued velocity potential on the whole exterior domain.
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