TITLE: Orientations on Manifold
QUESTION [5 upvotes]: This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example please help me ı want to learn orientation on manifold if ı could not understand in a good way this ı will not understand in a good way rest of the subject please help me

REPLY [6 votes]: Recall that an $n$-form associates to each point $p\in M$ a map $\omega_p:(T_pM)^n\to \mathbb R$. Thus for any basis $x_1,\ldots,x_n$ of $T_pM$, we have an associated real number $\omega_p(x_1,\ldots,x_n)$, which is either positive or negative. If we let $x_1,\ldots,x_n$ vary continuously with $p$ such that $x_1,\ldots,x_n$ is always a basis for $T_pM$, then $\omega_p(x_1,\ldots,x_n)$ is never $0$ and so must have constant sign. This sign gives you a notion of the orientation of the basis $x_1,\ldots,x_n$. In the case of the $1$-manifold $\mathbb R$, there is a non-vanishing $1$-form $dx$, and we have two possibilities up to sign for bases of $T_p\mathbb R$: either $1$ or $-1$, corresponding to left or right orientation. This generalizes to $\mathbb R^n$ in the way described by Sammy.