# Difference between vorticity and circulation

The definition of vorticity is $$\boldsymbol{\omega} = \nabla \times \mathbf{v}$$, 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, $$\boldsymbol{\omega} = \mathbf{0}$$.

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: $$C = \int_{\text{circular path}}{\mathbf{v}\cdot d\mathbf{l} = 2\pi ru_{\theta}},$$ a finite constant. But from the Stoke's theorem I know that: $$\int_{\text{enclosed area}}{\left(\nabla \times \mathbf{v}\right)\cdot d \mathbf{A}} = \int_{\text{enclosing curve}}{\mathbf{v}\cdot d\mathbf{l}},$$ 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?

Stokes' theorem cannot be applied here. The reason is that Stokes' theorem can only be applied to a surface that is enclosed by a simple, closed, piecewise-smooth curve (see any decent textbook on vector analysis). As Ben51 points out there is a singularity at the origin, so that must be excluded from the flow domain, which creates a hole in the domain. This means that the surface (the 'enclosed area' of your integration domain) is not enclosed by a simple closed curve, but it now lies between two curves, one is the outer enclosing curve and one is the curve which cuts out the origin. This means that we cannot apply Stokes theorem to compute the circulation, but we can of course compute it directly using line integration, $$\Gamma = \oint_C {\bf v} \cdot {\bf dr}$$.