Given the unit sphere $S^1$, and a 1 form $\alpha$ on $S^1$, Let $F:(0,2\pi)\to S^1$ be $F(\theta) = (cos \theta, sin \theta)$ be a parametrization of the sphere. Then we know that $\int_{S^1} \alpha = \int_{(0,2\pi)}F^{\ast}(\alpha)$ if $F$ is orientation preserving. If I let $S^1$ be oriented as the boundary of the unit ball, and the orientation on the unit ball is $dx \wedge dy$. My question is how to decide whether $F$ is orienation preserving or not.
This seems like a complicated process: Under the polar coordinate $F$ is just the identity, then I need to determine whether the polar coordinate is positively oriented with respect to $dx \wedge dy$. The only way to do this from my point of view is to write down the change of basis matrix, which involves arcsin. is that the correct way to do this?