Understanding $F \circ \phi^{-1}$ in differential geometry I am struggling with a question in elementary differential geometry. I thought I understood the basics until I read page 20 of The Geometry of Physics by T. Frankel. Suppose we have a manifold of dimension $m$. Furthermore, suppose we choose a chart $(U,\phi)$ and that we write $\phi$ explicitly as:$$y_j(\{x_i\})$$ where $i=1,...,m$ and $j=1,...,n$ and where $\{x_i\}$ is a coordinate system on $U\subset M$ and $\{y_i\}$ is in $\mathbb R$
Now here is my problem. Frankel says that if we have a function $F: M\rightarrow \mathbb R$, we can write it in terms of the $\{x_i\}$ as $$F_U:=F \circ \phi^{-1}$$ and he says that $F$ is a map $M \rightarrow \mathbb{R}$. 
I don't see this at all. I think that $F \circ \phi^{-1}$ maps $\mathbb {R^n} \rightarrow \mathbb R$. In other words, I think $F = F(\{y_j\})$ while he says $F=F(\{x_i\})$.
So my question is how to understand $F:=F \circ \phi^{-1}$ properly?
 A: Authors' definitions of chart can vary, but Frankel follows the convention that a chart $(U,\phi)$ on a manifold $M$ is a subset $U\subset M$ and a injective function $\phi:U\to \mathbb{R}^n$ whose image $\phi(U)\subset\mathbb{R}^n$ is an open subset of $\mathbb{R}^n$. Thus, $\phi^{-1}$ refers to the inverse function $\phi^{-1}:\phi(U)\to U$.
As to your question, Frankel is defining what it means to say that a function $F:M\to\mathbb{R}$ is differentiable as a function of a coordinate system $\{x^i\}$ on a patch $U$; specifically, that it is to be interpreted as meaning that $(F\circ \phi^{-1}):\phi(U)\to \mathbb{R}$ is a differentiable function in terms of the coordinates on $\phi(U)\subset \mathbb{R}^n$. 
So if you have coordinates $\{x^i\}$ on the patch $(U,\phi)$, coming from (or corresponding to) the coordinates $\{y^i\}$ on the open set $\phi(U)\subset\mathbb{R}^n$, we can write $F:M\to\mathbb{R}$ as a function of the $\{x^i\}$
$$F(x^1,\ldots,x^n)\text{ on the patch }U\subset M$$
then he says that "$F$ is a differentiable function of the coordinates $x$" is a statement which means
$$(F\circ \phi^{-1})(y^1,\ldots,y^n)\text{ on the set }\phi(U)\subset\mathbb{R}^n$$
is a differentiable function of the $\{y^i\}$.



