Is it possible to find a path from P to Q such that:
$$\int_C xy^2dx+ydy=0;\, P=(0,0)\, \text{and}\, Q=(1,1)$$
$$\int_C \frac{-ydx+xdy}{x^2+y^2}=0;\, P(-1,0)\, \text{and}\, Q=(1,0)$$
I've asked before for the possibility to find a path for vanishing in general, but now I want to know if there's a path for that particular points, Hints?
Do you know Green's Theorem? If so, you should be able to prove that the second one is impossible.
The first one is possible, but it's not so easy. Here's a hint. Take an easy path, say the straight line from $P$ to $Q$. Then follow that path by the path that goes around a rectangle from $(1,1)$ to $(a,1)$ to $(a,b)$ to $(1,b)$ and back to $(1,1)$ so that that integral cancels out the first one. (Green's Theorem will help here, too.)