Integrate ${xdy+ydx} \over {x^2+y^2}$ on the circle $(x-1)^2+(y-1)^2=1$

Question

I tried to integrate ${xdy+ydx} \over {x^2+y^2}$ on the circle $(x-1)^2+(y-1)^2=1$ counterclockwise. Used Grin's theorem, then went to polar cordinates but can't integrate the expression I got. So I am searching for another way.Need a hint where to look.

Answer 1

Integrate directly, by parametrizing.

I.e., go to a single-variable Riemann integral (from your line integral).

Compute your $dy$ and $dx$ terms like in Calculus I. And you should know how to proceed from here.

Answer 2

Green’s theorem is a good place to start. It says that $\oint_C P\,dx+Q\,dy=\iint_D\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\,dx\,dy$. What do those partial derivatives equal in this case? If you work that out before trying to convert to polar coordinates or other things, you might find that the double integral is particularly simple.