I am kinda stuck in this quite simple integral and I can't understand what I'm doing wrong. According to one of my uni's tests, the closed-path integral of $$F(x,y)=\Big(\frac{y}{2x^2+y^2}, \frac{-x}{2x^2+y^2}\Big)$$ along $\{(x,y)\in\mathbb{R}^2:2x^2+y^2=1\}$ is equal to $\sqrt2\pi$. However, I can't understand how is this possible, since $F(x,y)$ is irrotational and the set on which I'm supposed to integrate looks simply connected, to me, meaning that $F(x,y)$ should be conservative and therefore its circuitation around the given curve should be 0. What am I missing?
But even if it wasn't 0, I tried to integrate it in any way, and in all cases I get either stuck or 0. For instance, I tried with Green's theorem; however, the integrand is clearly 0 because $\frac{\partial F_2(x,y)}{\partial x}=\frac{\partial F_1(x,y)}{\partial y}$, assuming $F(x,y)=\big(F_1(x,y), F_2(x,y)\big)$. If I compute it with the help of a potential $U(x,y)$, I end up with $U(x,y)=\frac{1}{\sqrt2}\arctan\Big(\frac{\sqrt2x}{y}\Big)+C$, result that looks somehow more similiar to the result given by the test. However, I have no idea what endpoints to consider when calculating the difference, if I use cartesian coordinates, and I get undefined arguments of $\arctan$ when I use polar coordinates (like $\frac{\sqrt22\cos(2\pi)}{-\sin(2\pi)}$ when considering the ellipse running from $0$ to $2\pi$).
Could you give me an input on what's wrong with my computations? Thanks in advance for any help!
Your $F$ is not defined at $x=y=0$, so its domain is not simply connected.
The curve you're integrating over cannot be contracted to a point without passing through the origin.
Your idea of using a potential goes even worse. Then your expression for $U$ is not defined at $y=0$ at all. You could add $\pi$ to the arctangent for negative $x$ to make it possible to define a potential that is continuous on the positive $y$-axis, then it will not match up at all on the negative $y$-axis (or vice versa).