Intuition behind the orientation of arbitrary k-manifolds in arbitrary dimensions. I am wrestling with the intuition behind the definition of the orientation of a k-manifold in $R^n$.
The book that I am self-studying out of states that:
"Let M=X(D), where X: D $\epsilon$ $R^k$ $\Rightarrow$ $R^n$, be a smooth parametrized k-manifold. An orientation of M is a choice of a smooth, nonzero k-form $\Omega$ defined on M. If such a k-form exists, M is said to be orientable and orientated once a choice of such a k-form is made."
The book also states that:
"Let M=X(D), where X: D $\epsilon$ $R^k$ $\Rightarrow$ $R^n$, be a smooth parametrized k-manifold. The tangent vectors T$_{u1}$...T$_{uk}$ to the coordinate curves of M are said to be compatible with $\Omega$ if:
$\Omega_{X(u)}$(T$_{u1}$...T$_{uk}$)>$0$
We also say that the parametrization X is compatible with the orientation $\Omega$ if the corresponding tangent vectors T$_{u1}$...T$_{uk}$ are."
I know that this definition of orientability and $\Omega$ simplifies to defining the orientation of parametrized manifolds (k<2) using the directions of tangent and normal vectors. However, for k>2, is there any intuition behind this definition of orientability and $\Omega$?
(Let me know if the notation used in this question needs to be defined).
Thanks in advance.
 A: Maybe a topological point of view could be of use here.
A visual way to consider the orientation of any manifold is to imagine that you try to paint the manifold with two colors for each 'side'. More specifically, you consider the 2-fold orientation covering: locally you can orient as usual (as $\mathbf R^n$) and on the intersection you make orientations (colors) agree.
This 2-fold covering is connected if and only if the manifold is not orientable. Indeed, this means that there is a path that make you pass from the first color to the second. If the $2$-fold covering is not connected, then each color never meets the other: you can paint in blue all along a side without meeting the other side.
Now, from a differential point of view. You can still preserve this idea of painting. A $k$-form is a way to paint, with the colors $+$ and $-$. If $\omega$ is a local $k$-form then the space $\omega(X)>0$ can be distinguished from $\omega(X)<0$. Those are the two colors.
If you can find a global never vanishing $k$-form $\Omega$ then your manifold is oriented: take for example the color $\Omega(X)>0$ as in your book. Otherwise, there exists a path going from a color $-$ to $+$ meaning that any $k$-form will vanish.
