= Local spin in an azimuthal power-law flow
{title2=$\nabla\times(r^{1-2n}\hat\theta)=2(1-n)r^{-2n}\hat z$}
Away from the axis, the <cylindrical coordinates> formula for <curl> gives $r^{-1}\partial_r(rF_\theta)=2(1-n)r^{-2n}$ 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.
Back to article page
