Proposition 17.2 is showed as follows in Loring Tu's An Introduction to manifolds:
Proposition 17.2. If $f:M\longrightarrow\mathbb{R}$ is a $C^\infty$ funciton, then for $p\in M$ and $X_p\in T_pM$,
$$\left.f_*(X_p)=(df)_p(X_p)\frac{d}{dt}\right|_{f(p)}.$$
Proof. Since $f_*(X_p)\in T_{f(p)}\mathbb{R}$, there is a real number $a$ such that
$$\left.f_*(X_p)=a\frac{d}{dt}\right|_{f(p)}.\tag{17.1}$$
To evaluate $a$, apply both sides of $(17.1)$ to $x$:
$$a=f_*(X_p)(t)=X_p(t\circ f)=X_pf=(df)_p(X_p).\qquad\square$$
I cannot understand why $X_p(t\circ f)=X_p f$. Can anyone clarify this for me?
Best Answer
Here, $t\colon \Bbb R \to \Bbb R$ is the global coordinate function on $\Bbb R$. The first $\Bbb R$ plays the role of the "manifold", the second $\Bbb R$ plays the role of the Euclidean space modelling the manifold. Compare: isn't a chart $(U,\varphi)$ for a manifold $M$ a homeomorphism $\varphi\colon U \subseteq M \to \varphi[U]\subseteq \Bbb R^n$? Here, $U = M = \Bbb R$, $\varphi$ is called "$t$", and $n=1$. So given $p \in M$, we have that $f(p) \in \Bbb R$ (the manifold $\Bbb R$). What is the coordinate of this point $f(p)$ in the manifold $\Bbb R$? Apply the chart $t$ to get $t(f(p)) \in \Bbb R$ (the Euclidean space $\Bbb R$). What makes this example so particular (and caused your confusion) is that $t(f(p)) = f(p)$ (numerically).