This is a follow up to this question, in which the context is provided. I would like to see the equivalence of two notions of tangent spaces and differentials of smooth maps.
Guillemin and Pollack define the tangent space as follows. Let $\phi: U \subseteq \mathbb{R}^n \to M\subseteq \mathbb{R}^N$ be a coordinate map for the embedded submanifold $M$ s.t.\ $0 \in U$ and $\phi(0) = p$. Then $D\phi|_0$ is the derivative of $\phi$ at $0$. Then let $T_pM := D\phi|_0( \mathbb{R}^n)$ i.e. a certain $n$ dimensional vector subspace of $\mathbb{R}^N$. It is then proved that this definition is independent of $\phi$.
Tu defines the tangent space of the vector space of derivations of germs of $C^\infty$ functions at a point $p$.
How do these definitions correspond?
Under this correspondence, we should get a correspondence between notions of the differential of a smooth map. Let $M, N \subseteq \mathbb{R}^N$ be $m, n$ resp. dimensional embedded submanifolds. Let $f: M \to N$ be a smooth map between the two manifolds. From this we want to construct a vector space homomorphism from $T_pM \to T_{f(p)}N$ for any $p \in M$. Tu and G&P do this in two different ways.
G&P define $df_p$ as follows. Let $\phi: U \to M$ and $\psi: V \to N$, where $U,V$ are open subsets of $\mathbb{R}^m, \mathbb{R}^n$ resp., be coordinate maps centered at $p$ and $f(p)$ resp. Then let $h = \psi^{-1} \circ f \circ \phi: U \cap f^{-1}(\psi(V)) \to \mathbb{R}^n$. Then $h$ is smooth and we define $df_p = d\psi_0 \circ dh_0 \circ d\phi^{-1}_0$, where $d\psi_0$ is just the total derivative of $\psi$ at $0$. It is then verified this definition is in fact coordinate independent.
Tu defines the differential of $f$ by defining its action on $C^\infty$ real-valued functions (as this will uniquely determine a derivation): $f_*(v)(g) =v(g\circ f)$.
How do these definitions correspond?
Best Answer
I've written up a somewhat lengthy but very explicit demonstration of the correspondence between Tu's and G&P's definitions of smooth manifolds, smooth maps, tangent spaces, and differentials. I will leave a link to the pdf of the write up here for anyone who might want it:
https://drive.google.com/file/d/1WvfwaKvKLp13Swioqy0ElLb7zLxisi_D/view?usp=sharing