I was asked the following question:
Given the following vector fields $$ G(x,y)=(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2}),$ $$ $$ H(x,y)=a G(x,y-1) + bG(x,y+1), $$ and the path defined by $$ r=\{\ (x,y) \in \mathbb{R}^2: ^2 + (\frac{y}{4})^2=1 \wedge y\ge0 \}, $$ calculate the work done on the path from $(-1,0)$ to $(1,0)$ along $r$ by the vector field $H$.
On the previous exercises, it was already proven that $H(x,y)$ is closed and the work done on the line that goes from $(-1,0)$ to $(1,0)$ is given by $\frac{\pi (a-b)}{2}$
At first I thought that I could apply Green's theorem, since the vector field is closed I would get $0$ and then I would just need to subtract the work done on the line. But for some reason, when I checked my resolution with my teacher's, the results obtained were different.
The way he solved involved using some kind a"homotopic" argument, which I didn't understand where he calculated the work done on a circunference radius 1 around $(0,1), (cos(\theta),1+sin(\theta))$.
I presumed that I couldn't use Green's Theorem but why? And secondly what is this "homotopic" argument and how to use it to solve this kind o problems.