Let $\gamma$ be the oriented piecewise $\mathcal C^1$-arc consisting of the line segment from $(1,0)$ to $(0,1)$, followed by the line segment from $(0,1)$ to $(-1,0)$. What is $$\int_\gamma \frac{x\,dy-y\,dx}{x^2+y^2}?$$
I am trying to parametrize this arc so that I can integrate the $1$-form but I am having trouble doing this. Any hints?
Hint: Use Green's Theorem to relate the integral over the union of two line segments to an integral over a half-circle.
Two reasons: 1. It is easy to compute the integral over the upper half unit circle, and 2. The result is the same as computing the integral over the union of two line segments.
Use Green's Theorem on the closed curve consisting of the line segment from $(1,0)$ to $(0,1)$, followed by the line segment from $(0,1)$ to $(-1,0)$, then the upper half circle centered at the origin from $(-1,0)$ to $(1,0)$ - caveat: the curve above is oriented clockwise, but the result is 0.