I am currently reading a chapter about Pfaff forms, but not really understand, why the author introduces tangential vector spaces - the definition seems rather redundant to me, if I didn't overlook something.
Let $ U \subset \mathbb{R}^n$ open, and $ p \in U$ then: $T_p (U) $ is called the tangential vector space in point $p$ - the set of all tangent vectors $\alpha ' (0)$ cont. diff able through point p,
$\alpha : (-\epsilon, \epsilon) \to U $ and $ \alpha(0) = p $
This implicates: $ T_p(U) = \mathbb{R}^n $ as: $\forall v \in \mathbb{R}^n :t \to p +tv $ is a tangential vector
So why is this vector space needed?
Directly after that the dual space $T_p^* (U) $ to $T_p(U)$ is defined (which is used & useful as the following chapters tell) - however I don't really see the idea / the need to go the way over the tangential space, when it is equal to $\mathbb{R}^n$ .
I am afraid I miss out an key idea but this tangential space and if I don't get it, then I won't understand all following definitions well and overall miss a lot.
Thus I would be very happy, if someone understands my confusion and may clear things up. As always I am happy about any constructive comment or answer.
The idea makes more sense if you think about surfaces.
For example take $U \subseteq \mathbb R^2$ open and some sufficiently nice function $f: U \to \mathbb R$. Then, the graph $G_f = \{(x,y,f(x,y))\}$ of $f$ is a surface in $\mathbb R^3$ and one can consider the tangential vector space to a point $p \in G_f$ which would then be defined to
$$T_p(G_f) = \{ \alpha'(0) | \alpha: (-\varepsilon , \varepsilon) \to G_f \text{ cont. diff. } , \alpha(0) = p \}$$
In this case $T_p(G_f)$ (or rather $T_p(G_f) + p$) can really be interpreted as a plane which is tangential to $G_f$ at $p$.