Calculate $\oint_{x^2+y^2=6} Pdx+Qdy$ Given the vector field $F(\frac{x+y}{x^2+y^2},\frac{-x+y}{x^2+y^2})$
I checked if the the field is Conservative and got $P_y=\frac{x^2-y^2-2xy}{(x^2+y^2)^2}$ and $Q_x=\frac{x^2-y^2-2xy}{(x^2+y^2)^2}$ so the field is conservative and it satisfies $Q_x=P_y$ but our field is not a simple connected space because the field is not defined at $(0,0)$ and our circle $x^2+y^2=6$ includes this point . and from what I know in order to use green's theorem we need a simple closed curve and a counterclockwise direction and an area inside of the curve so according to that I cannot use green's theorem here.
but in the book they solved it by simply applying the information on the line integral and got $-2\pi$ but in another chapter in the book it states that if the field is conservative then $\oint F \cdot dr=0$
I searched and found on the book a chapter "green's theorem in multiply connected region" I understood that I can take a simpler circle and integrate it and I will get the same answer. but we did not learn that so I wont use it and the final answer ($-2\pi$) is least important here.
My question is why isn't the integral equal to zero? and why they calculated the line integral normally without taking in consideration that if the field is conservative then $\oint F \cdot dr=0$ ?
Thank you and sorry for the English mistakes