# Difference between vorticity and circulation

The definition of vorticity is $\mathbf{\omega} = \nabla \times \mathbf{v}$, where $\mathbf{v}$ is the velocity vector field.

Now, if I look at a rotating flow in cylindrical coordinates I find that: $$\nabla \times \mathbf{v} = \frac{1}{r}\frac{\partial (r v_{\theta})}{\partial r},$$ in case of a free vortex I also know that $v_{theta} \propto 1/r$ and therefore the derivative in the above equation vanishes for a vortex centered at $r=0$. In other words, the vorticity is zero everywhere, $\mathbf{\omega} = \mathbf{0}$.

I can also look at the situation globally, and instead of the localised curl I take a line integral of the speed along a circular path distance $r$ from the centre. In this case I find that: $$C = \int_{\textit{circular path}}{\mathbf{v}\cdot d\mathbf{l} = 2\pi ru_{\theta}},$$ a finite constant. But from the Stoke's theorem I know that: $$\int_{\textit{enclosed area}}{\left(\nabla \times \mathbf{v}\right)\cdot d \mathbf{A}} = \int_{\textit{enclosing curve}}{\mathbf{v}\cdot d\mathbf{l}},$$ but if the circulation is a finite non-zero constant, the curl must be also non-zero somewhere within the enclosed area! Thus the vorticity is non-zero somewhere in the velocity vector field!

These two findings seemingly contradict each other, where am I making a mistake?