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I am reading introduction to smooth manifolds by John M. Lee. I don't understand a part of change of coordinates.

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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.

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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.

Tian Vlašić
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romperextremeabuser
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    The term coordinate function is defined by the author on page 4 in the book you are reading (start of the section titled "Coordinate Charts"). – jd27 Aug 06 '23 at 10:48
  • Not really. It says that $\varphi$ is a (local) coordinate map (I assume map=function), but In the picture I sent it is written: "denote coordinate functions of $\varphi$ as $x^i$". So maybe coordinates functions of $\varphi$ are coordinate funtions, but then If $x=(x^1,x^2,...,x^n)$ it make no sense to me, they are treated as functions and numbers at the same time. Maybe $x$ is not equal to (x^1,x^2,...,x^n) but it is on the second picture I sent. – romperextremeabuser Aug 06 '23 at 17:28
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    See https://math.stackexchange.com/q/4706005. – Paul Frost Aug 06 '23 at 22:58

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