This problem focuses on the distinction between local irrotationality and global circulation in a non-simply connected domain. Although the local curl of the vector field $v$ is zero everywhere except at the $z$-axis (where $\rho=0$ ), the field exhibits a "vortex" nature that results in a non-zero circulation when integrated along a path that encloses that singularity. This specific vector field behaves similarly to the magnetic field around a current-carrying wire; it is conservative locally but not globally. Because the curve $\Gamma$ winds around the $z$-axis twice (as indicated by $\phi$ ranging from 0 to $4 \pi$ ), the line integral yields $4 \pi$-a result that confirms Stokes' Theorem cannot be applied across a surface that intersects the $\rho=0$ singularity without accounting for the singular behavior at the origin.
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%% Condensed Notes
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%% Proof and Derivation
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