I am reading introduction to smooth manifolds by John M. Lee. I don't understand a part of change of coordinates.
What are $(x^i)$ and $\tilde{x}^i$. It says here that these are "coordinate functions". What are coordinate functions? In this post What is a coordinate function $x^i$ of a manifold, given a chart $(U,x)$? it says that $i$-th coordinate function output $i$-th coordinate of point in $R^n$. Is it true? Is $x=(x^1,x^2,...,x^n)$ and $\psi \circ \varphi^{-1}(x)=\psi \circ \varphi^{-1}(x^1,x^2,...,x^n)$?. If yes (but I don't think so because these are treated like number, elements from $R^n$, not functions just by looking at the domain of the map) then I can't understand following picture.
The first $n$ coordinates of $\psi \circ \phi^{-1}(x^1,x^2,...,x^n,v^1,v^2,...,v^n)=(\tilde{x}^1(x),\tilde{x}^2(x),...,\tilde{x}^n(x),...)=(\tilde{x}^1(x^1,x^2,...,x^n),\tilde{x}^2(x^1,x^2,...,x^n),...,\tilde{x}^n(x^1,x^2,...,x^n),...)$
But now these $x^i$ are numbers, elements of $R^n$, not functions. I don't understand what is going on.