I have the result that any 1 dim topological manifold is either $\mathbb{R}$ or $S^{1}$. And I have the fact that every 1-dim topological manifold is orientable in the sense of orientation on simplices.
i want to get that any 1-dim manifold (smooth) is orientable, where orientability is given by the existence of a nowhere vanishing 1-form. Since i know my manifold is either the real line or the circle, does the section $ s: M \longrightarrow T^{*}M$ that takes each point p to the differential dx at p work as the 1-form i need?
