Zero curl but nonzero circulation

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SUMMARY

The vector field \(\vec{F} = \left<\frac{-y}{x^2 + y^2},\frac{x}{x^2 + y^2},0\right>\) exhibits zero curl, indicating that its circulation is zero. However, the integral \(\int \vec{F} \cdot d\vec{s}\) around a unit circle in the xy-plane yields a non-zero result of \( \pm 2\pi\). This discrepancy arises because \(\vec{F}\) is undefined at the origin (0,0), leading to a non-simply connected domain. Consequently, Stokes' theorem cannot be applied as the required surface does not exist without including the singularity at the origin.

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The vector field \vec{F} = &lt;\frac{-y}{x^2 + y^2},\frac{x}{x^2 + y^2},0&gt; has a zero curl, which means its circulation is zero. However

\int \vec{F}.d\vec{s} around a unit circle on the xy plane is equal to (+/-)2\pi and not zero

Is it because F is undefined at (0,0)? No, because Stoke's theorem allows me to choose an arbitrary surface not including the origin(0, 0, 0)?
 
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Is it because F is undefined at (0,0)?
Right - more general, it is undefined at (0,0,z). As result, the domain of F is not simply connected - and the surface you want to consider does not exist.
 

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