I wish to calculate the following integral using Green's theorem:
where $\gamma$ is the unit circle.
If I understand correctly, I can't use Green's theorem since the function is not $C^1$ at $(0,0)$. so the field is locally conservative but not around that point.
The idea is to use the following:
Since in the yellow area the field is conservative, the integral is $0$. the two paths that lead to the inner circle cancel out each other. The last thing I need to show is that for any field around the point $(0,0)$ the integral will be $2\pi$. How can I show that? and why does it work for every field?