Rank and Jacobian matrix of smooth $F:M\to N$ between manifolds I'm currently reading about submersions and immersion's Lee's Introduction to Smooth Manifolds (p.77), and I'm slightly confused about what is meant when he says that the rank of $F$ and $p$ is "the rank of the Jacobian matrix of $F$ in any smooth chart."
Letting $(U,\varphi)$ and $(V,\psi)$ be the local charts at $p$ and $F(p)$, respectively, I was wondering if the Jacobian matrix that was mentioned referred to the Jacobian matrix of $\psi\circ f\circ\varphi^{-1}$. This matrix seems to make sense because $\psi\circ f\circ\varphi^{-1}$ maps between Euclidean spaces, but I was wondering if the "Jacobian matrix of $F$" could be defined in a way that is independent of charts (if this makes any sense).
 A: Given your map $f: M \rightarrow N$ (where $\dim M = m$ and $\dim N = n$), the rank of the Jacobian matrix at a point $p\in M$ is independent of your choice of charts, which is why Lee defines it to be the rank of the differential $df_p : T_pM \rightarrow T_{f(p)}N$, a linear map that does not exist as a matrix until you assign a basis on the tangent spaces. In practice, actually computing the rank of $df_p$ requires you to assign a coordinate chart (over the sets $U \subset M$ and $V \subset N$ let's say) as you've indicated.
The reason rank is independent of the chart chosen is because $\phi : U \rightarrow \mathbb{R}^m$ and $\psi : V \rightarrow \mathbb{R}^n$ are diffeomorphisms, and hence their differentials are isomorphisms on the level of tangent spaces. You should convince yourself then that for purely linear algebraic reasons:
$$\text{rank }df_p = \text{rank } d\psi\circ df_p \circ d\phi^{-1} = \text{rank } d(\psi\circ f \circ \phi^{-1})_p.$$
I hope that makes it a little clearer for you!
A: He defined the rank of $F$ at $p$ to be the rank of the linear transformation $dF_p:T_pM \to T_{F(p)}N$ (where the rank of a linear transformation is say defined as the dimension of the image). Next, what he's claiming is that for every chart $(U,\phi)$ about $p$ and $(V,\psi)$ about $F(p)$, we can consider the linear transformation $D(\psi\circ F \circ \phi^{-1})_{\phi(p)}:\Bbb{R}^{\dim M} \to \Bbb{R}^{\dim N}$ (this is the usual derivative at a point for a map between open subsets of a finite-dimensional normed vector space). THe claim is then that
\begin{align}
\text{rank} \left(dF_p\right) &= \text{rank} \left( D(\psi\circ F \circ \phi^{-1})_{\phi(p)}\right) \\
&= \text{rank} \left( (\psi\circ F \circ \phi^{-1})'(\phi(p))\right)
\end{align}
where $(\psi\circ F \circ \phi^{-1})'(\phi(p))$ is the matrix representation of the linear transformation $D(\psi\circ F \circ \phi^{-1})_{\phi(p)}$ with respect to the standard ordered bases for $\Bbb{R}^{\dim M}$ and $\Bbb{R}^{\dim N}$, i.e the Jacobian matrix in a chart (the second equality about rank of linear transformation vs rank of matrix representation being equal is a standard linear algebra fact).
The notion of the differential/push-forward/tangent-mapping $dF_p:T_pM \to T_{F(p)}N$ is exactly meant to generalize the concept of "Jacobian matrix" (actually even for maps between open subsets of Euclidean spaces, we can consider the derivative at a point as a linear transformation, and then the Jacobian matrix is just the matrix representation of this linear map relative to the standard bases). See the remarks made on page 62-63.
