In $\mathbb{R}^2$ consider $\gamma_0$ as the circumference centered in $(0, 0)$ with radius 1 and $\gamma_1$ as the circumference centered in $(1, 1)$ with radius 4. I'm looking for a differential form $\omega$ defined in $\mathbb{R}^2-\{(0, 0), (1, 1)\}$ such that:
$$\int_{\gamma_0}{\omega}=1 \ \ \ \ \ \ \ \ \ \int_{\gamma_1}{\omega}=-1$$
I also want that form to be $\mathscr{C}^1$ where it's defined.
I only understood that the form cannot be exact but I don't how to continue, please keep the example as simple as possible.