$ \int_\Gamma \frac{x \; dy-y \; dx}{x^2+y^2} $ where $\Gamma$ is a circle: $x^2+y^2-2x-2y+1=0$
By "completing the square" I see that the circle has a radius of $1$ and is moved one point to the right and one up. Partial derivatives of $M(x,y)$ and $N(x,y)$ are: $ \frac{\partial M}{\partial y}=\frac{y^2-x^2}{(x^2+y^2)^2} $ $ \frac{\partial N}{\partial x}=\frac{y^2-x^2}{(x^2+y^2)^2} $ So, I get $ \int\int \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) \; dA = \int\int dA $ Am I correct up to this point? How do I continue? What should be the bounds for r if I convert it into polar coordinates?