Let us define the map $f:\mathbb{R}\to \mathbb{S}^1$ by $x\to e^{2\pi i x}$. I want to prove that this map is an orientation-preserving map and also it's restriction on the unit interval has an orientation-preserving inverse.
What I know:
$f: M\to N$ is an orientation preserving if the determinant of the derivative map on the tangent space level is positive. So here, we need to show $Df(p):T_p\mathbb{R}\to T_{f(p)}\mathbb{S}^1$ has a positive determinant. But here $Df(p)$ is just a real number so we need to show it is a positive number. Consider a curve $\gamma(t)=p+tv$ where $(p,v)\in T_p\mathbb{R}$. $f(\gamma(t))=e^{2\pi i(p+tv)}$. But here I am having difficulty as the derivative will be a $2\times 1$ matrix.