What is the difference between Jacobian of maps between manifolds and matrix representation of its differential?

Question

In An Introduction to Manifolds by Loring Tu p.67 and 91, the author first defined the "Jacobian matrix" (see the picture) of maps between manifolds relative to corresponding charts. And then in p.91, he discuss local expression of differential (which is a linear transformation between tangent spaces) and discuss its matrix representation. However, I want to ask is these two things literally the same?

Answer 1

If you have a map $ f : U \to V$ between open subsets $U \subset \mathbb R^m, V \subset \mathbb R^n$, then the derivative at $p \in U$ is the best linear approximation of $f$ in $p$. This is the (unique) linear map $df(p) : \mathbb R^m \to \mathbb R^n$ such that $$\lim_{h \to 0} \dfrac{\lVert f(p+h) - (f(p) + df(p)(h)) \rVert}{\lVert h \rVert} = 0. $$ Note that for the above quotient to be well-defined it is essential that $U \subset \mathbb R^m$ (so that $p + h$ is defined) and $V \subset \mathbb R^n$ (so that $f(p+h) - (f(p) + df(p)(h))$ is defined).

If we have a map $F : M \to N$ between general manifolds, we obviously cannot define the derivative of $F$ at $p \in M$ as above: Neither there exists a sum $p + h \in M$ for $p \in M$ and (sufficiently small) $h \in \mathbb R^m$ nor the expression $f(p+h) - (f(p) + df(p)(h))$ makes any sense.

What we can do is to choose coordinate systems $(U,\varphi)$ at $p \in M$ and $(V,\psi)$ at $F(p) \in N$ such that $F(U) \subset V$. Then $F_{\varphi,\psi} = \psi \circ F \circ \varphi^{-1}$ is a map between open subsets of Euclidean spaces which has a derivative $d_{\varphi,\psi}F(p) = dF_{\varphi,\psi}(\varphi^{-1}(p)) : \mathbb R^m \to \mathbb R^n$. This depends on the coice of the coordinate systems, i.e. this approach does not produce a derivative of $F$ in an absolute sense. However, the canonical matrix representation of $d_{\varphi,\psi}F(p)$ is the Jacobian matrix of $F_{\varphi,\psi}$.

The tangential space $T_p M$ of a manifold $M$ at $p \in M$ is defined without reference to any coordinate system at $p$. Howewer, any choice of a coordinate system $(U,\varphi)$ at $p$ yields a specific linear isomorphism $\iota_\varphi : \mathbb R^m \to T_pM$. The derivative $F_{*,p} : T_pM \to T_{F(p)]}N$ can be defined without reference to coordinate systems - it has a coordinate-free meaning. This is an essential benefit of working with tangential spaces.

Proposition 8.11 describes the relation between betwen the coordinate-free concept of derivative $F_{*,p}$ and the "classical" derivatives $d_{\varphi,\psi}F(p)$ depending on the choice of coordinates. In fact, we have a commutative diagram

$\require{AMScd}$ \begin{CD} \mathbb R^m @>{d_{\varphi,\psi}F(p)}>> \mathbb R^n \\ @V{\iota_\varphi }VV @V{\iota_\psi }VV \\ T_pM @>{F_{*,p}}>> T_{F(p)}N \end{CD}

in which the vertical arrows are isomorphisms depending on $\varphi,\psi$.