Let $\vec{F}=(P,Q) \in C^1 (\mathbb{R}^2 - (0,0) ) $ such that $Q_x=P_y $ and let $\gamma$ be a closed line surrounding the origin such that $\int_{\gamma} \vec{F} \cdot \vec{dr} =0 $ . Show that $\vec{F}$ is conservative in $\mathbb{R}^2 -(0,0)$ .
I know I need to show this result by proving that the line integral over every closed line vanishes. So, let $ C $ be such a contour. I know how to prove this for $C$ that is contained in $\gamma$ and for $C$ that contains $\gamma$. But how can I prove this for the case $\gamma$ and $C $ intersect but one is not contained in the other one?
Thanks a lot !
