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Definitions and notation

Definition of a differential in an open set

Let $U\subset \mathbb R^n$ an open set. We define the differential form of degree $r$ $\omega$ in the open set $U$ as the function $x\in U\mapsto\omega(x)\in A_r(\mathbb R^n)$.

Definition of a differential in an manifold

Let $M$ a manifold. We define the differential form of degree $r$ $\omega$ in the manifold $M$ as the function $x\in M\mapsto\omega(x)\in A_r(T_xM)$.

Notation

$T_xM$ is the tangent vector space at the point $x\in M$ and $A_r(E)$ is the vector space of the alternating $r$ forms of a vector space $E$.

My doubt

What are the relation of these definitions with each other? Where do they overlap? I've heard that we can see locally the differential of a manifold as the differential of an open set (I suppose a parametrized one). The problem is a generic point $\omega(x)\in A_r(\mathbb R^n)$ not necessarily is in $A_r(T_xM)$.

Please I really need help, I've being trying to understand this for days. (See for example my last question and the comments). I would be really grateful if someone could help me to understand this so that I can go on my studies.

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    In the first definition, it might be better to say $A_r(T\Bbb{R}^n)$ instead of $A_r(\Bbb{R}^n)$. You use the chart maps/parameterizations to identify open sets in the manifold with open sets in $\Bbb{R}^n$, and identify $TM$ with $T\Bbb{R}^n$ (locally). – 2017-02-28
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    @Nick I'm sorry I"m really dumb on this subject. Could you please give more details how to make this identification $T_xM$ with $T_x\mathbb R^n$ locally? are you seeing $\mathbb R^n$ as a manifold? Thank you for your comment. – 2017-03-04

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Around each point $p \in M$, you have some neighborhood $U$, and local coordinates $(x_1,\dots,x_n)$ on $U$. You think of this as a "chart" with a diffeomorphism $\varphi \colon U \to V$, where $V$ is some open set in $\Bbb{R}^n$ containing the origin with standard coordinates $(x_1,\dots,x_n)$, and $\varphi(p) = (0,\dots,0)$.

Now the "tangent map" $d\varphi \colon T_pM \to T_0\Bbb{R}^n$ is an isomorphism of vector spaces, since $\varphi$ is a diffeomorphism. This is what I meant above in the comments when I said we identify $TM$ and $T\Bbb{R}^n$ locally.

A global differential form $\omega$ on $M$ restricts to an alternating multilinear map on each tangent space $T_pM$. Using the identification above, we use the coordinate chart to identify this with a multilinear form on $T_0\Bbb{R}^n$ in the standard coordinates.

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    What is your definition of $TM$? The book I'm reading only defines $T_xM$ applied at a point. Thank you very much for your answer – 2017-03-04
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    $TM$ is the tangent bundle. It is the vector bundle over $M$ which is the disjoint union of all the tangent spaces. – 2017-03-04
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    In the book I'm studying the parametrizations $\varphi$ are homeomorphisms and immersions, can we say they are diffeomorphic in this context as well? thank you again. – 2017-03-04