I have to integrate this line integral $$\int_C \mathbf F \cdot d\mathbf r$$
Where $\mathbf F = (\frac{y}{x^2 + y^2},\frac{-x}{x^2 + y^2})$ and $C$ is the curve $x^2 + 2y^2 = 1$ oriented anticlockwise
Using parametrisation the answer is $-2\pi$ which is correct but does not use a vector theorem. I tried using green's theorem but the integral becomes $0$ after taking the necessary derivatives which comes from the discontinuity at the origin? Is there a way to apply greens theorem in this case, or is there another theorem that could be used? Thanks