Local spin in an azimuthal power-law flow ($\nabla \times (r^{1-2n}\hat{\theta })=2(1-n)r^{-2n}\hat{z}$)

Away from the axis, the cylindrical coordinates formula for curl gives $r^{-1}{\partial }_r(r{F}_{\theta })=2(1-n)r^{-2n}$ when $F_{\theta }=r^{1-2n}$. Its sign changes at $n=1$, even though the tangential velocity stays positive. Thus local angular velocity, one half of vorticity, can oppose orbital circulation. For $n=1$, the field is a line vortex with zero ordinary curl off the axis and circulation $2\pi$ around it. Stokes theorem applies on annuli excluding the axis with both boundary components included; it cannot be applied to a spanning disc containing the field singularity.