Zero curl but nonzero circulation

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The vector field F = <(-y)/(x^2 + y^2), (x)/(x^2 + y^2), 0> exhibits zero curl, indicating zero circulation under typical conditions. However, the circulation integral around a unit circle in the xy plane yields a nonzero value of ±2π, raising questions about the implications of the field's behavior at the origin (0,0). The discussion clarifies that the field is undefined at (0,0) and is not simply connected, which affects the applicability of Stoke's theorem. It emphasizes that the chosen surface for the integral must exclude the singularity at the origin. This highlights the complexities in vector fields with singularities and their impact on circulation calculations.
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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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