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I’m currently reading “Fundamentals Of Aerodynamics” by John D. Anderson.

Going through discussions on some elementary flows, I encountered Vortex flow which according to the author is irrotational everywhere except at the centre where its infinite. enter image description here

Upon calculating the vorticity, however, I found it to be rotational everywhere, being inversely proportional to the square of the distance from the centre.

So, where am I going wrong ?

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Your expression for the curl is wrong. You're working the curl out with the determinant mnemonic that only works for Cartesians. Analogous determinant expressions exist for other co-ordinate systems, but you need to look them up. You'll find that, in particular, the $z$-component is:

$$\frac{1}{r}\left(\frac{\partial}{\partial\,r}(r\,A_\theta)-\frac{\partial}{\partial\,\theta}(A_r)\right)$$

which vanishes aside from at the singular point at $r=0$.

The fundamental way to think about the curl is as the operator that, given specification of a plane, returns the circulation around a loop in that plane per unit area, in the limit as the loop's enclosed area tends to nought. That process gives you the extra $r$ factor that annuls the $1/r$ in your expression.

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