Untidy notations in differential geometry is killing me. There are many instances of this but I really need a help to clarify this one in particular.
Suppose $M$ is a manifold with a coordinate chart $x=(x_1,\cdots x_n):M\rightarrow \mathbb R^n$. Often people talk of smooth function $f(x_1,\cdots x_n)$. Here I find myself quite confused; if each component of a coordinate chart is by definition a smooth function on $M$, how can you consider these as arguments of another function $f$? A similar situation arises when $M$ is a Riemann surface with a coordinate chart $z:M\rightarrow \mathbb C$, and for smooth function $f$ we make a power series expansion $f(z)=a_0+a_1z+a_2z^2+\cdots$ which is equally as weird.
My understanding of this is that once we write $f(x_1 \cdots x_n)$, $f$ no longer literally means $f$ and instead means $f\circ x^{-1}$, and that $x_1 \cdots x_n$ taken as arguments of $f$ are no longer mean the coordinate functions and are actually instead dummy variables waiting for real numbers to be plugged in. However I am not all that sure. Please tell me if I am right or wrong, and if I am wrong, please tell me the correct way to make sense of this.
You are right, what is done to the function is called pullback.
http://en.wikipedia.org/wiki/Pullback_%28differential_geometry%29#Pullback_of_smooth_functions_and_smooth_maps
In this case you have as map $x^{-1}:\mathbb{R}^{n}\rightarrow M$.