Question 9.12.5: Region with a Hole Evaluate ∮C -y/x²+y² dx + x/x²+y² dy, whe......

Because P(x, y) = \frac{-y}{x^{2}+y^{2}}, Q(x, y) = \frac{x}{x^{2}+y^{2}}, and the partial derivatives

\frac{\partial P}{\partial y} = \frac{y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}, = \frac{\partial Q}{\partial x} = \frac{y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}

are continuous on the region R bounded by C, it follows from (4) that

\oint_{C} \frac{-y}{x^{2}+y^{2}} d x+\frac{x}{x^{2}+y^{2}} d y = \iint_{R}\left[\frac{y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}-\frac{y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}\right] d A = 0 .