I am reading Methods of Information Geometry by " Shun-Ichi Amari",
I got stuck in the following line that I have underlined in the image as you can see.
My question is we have a real-valued function $f: S\to \mathbb{R}$ and we have $[\xi^i]$ to be the global coordinate of $S$, we have $\varphi: S\to \varphi(S)\subseteq \mathbb{R^n}$ so now we have the composition $f\circ\varphi^{-1}:\varphi(S)\to \mathbb{R}$ which the author is representing by $\bar{f}=f\circ \varphi^{-1}$, I am a bit confused in the partial derivative of $\bar{f}$ since it is a function from $\varphi(S)\subseteq \mathbb{R^n}\to \mathbb{R}$ so according to me we should differentiate it with respect to the coordinate of $\mathbb{R^n}$. Still, here they are differentiating with respect to the coordinates of $S$? what am I missing here? In my opinion, if we let $u^{i}$ be the coordinate of $\mathbb{R^n}$ then $\frac{\partial f\circ\varphi}{\partial u^i}$.So I guess the author is also taking $[\xi^i]$ as a coordinate of $\varphi$? is it?
You are right, notation is a bit sloppy. The author considers a chart $\varphi: U \to V \subset \mathbb R^n$ on $S$ and writes it in the form $\varphi = [\xi^i]$, where the $\xi^i : U \to \mathbb R$ are the coordinate functions of $\varphi$. At the same time he also writes $\bar f(\xi^1,\ldots,\xi^n)$ which suggests that these $\xi^i$ are the $n$ coordinate variables of $\bar f$.
This is in fact ambiguous: $\xi^i$ denotes both a coordinate function (which needs an input $s \in U$ to produce a real number $\xi^i(s)$) and a variable of the function $\bar f : V \to \mathbb R^n$.
As you I would prefer to use other symbols like $u^i$ for the variables of $\bar f$. Anyway, if you understand the notational ambiguity, it should not be a problem to correctly interpret $\frac{\partial \bar f}{\partial \xi^i}$. And be aware that the variable names used as arguments for $\bar f$ do not play any role. If the $i$-variable has the name $n$, then we write $\frac{\partial \bar f}{\partial n}$ for the $i$-th partial derivative of $\bar f$. This leads to notations like $\frac{\partial \bar f}{\partial x^i}$ (which seems to be the most popular variant), $\frac{\partial \bar f}{\partial u^i}$, $\frac{\partial \bar f}{\partial y}$ etc. for the same thing. Concerning the use of variable names also have a look to Difficulty understanding Chain rule in multivariable calculus.
Here is a similar question which might be worth studying.