Given a fluid element whose circulation was determined counter-clockwise, how can I calculate the Vorticity of the element?
The circulation is $$\Gamma = (\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2})\delta x_1 \delta x_2$$ I thought this might give a better idea about the question. Here $u_1$ is the velocity along $x_1$ and $u_2$ is the velocity along $x_2$.
The circulation relation you wrote is exactly the circulation for a fluid particle. The term between parenthesis is then the vorticity of this particle.
I don't think one could possibly have an answer to your question. Circulation is a derived quantity and essentially depends on the choice of the closed curve and the flow field across that curve. Thus leaving you with to variables to deal with (in fact 3 variables, the third one being ; establishing the relation between the velocity field and the choice of the curve). So the question why have Circulation...The quantity like momentum remains conserved unless acted upon by baroclinic effects, non conservative forces (coriolis and friction). Thus making the physical understanding of the physical situation easier. One does not have these benefits when dealing with the vorticity equation. Do let me know if you would want to discuss this any further...
