Can anyone explain the meaning of this excerpt from Tu's Introduction to Manifolds? I haven't ever seen this notation used before. I also provide an attempt at understanding what he wrote.
In the context of manifolds, we denote the standard coordinates on $\mathbb R^n$ by $r^1 , \dots , r^n$
If $(U, U \xrightarrow{\quad \varphi \quad} \mathbb{R}^n)$ is a chart of a manifold, we let $x^i = r^i \circ \varphi$ be the $i$th component of $\varphi$ and write $\varphi = (x^1 , \dots , x^n)$ and $(U, \varphi) = (U, x^1, \dots , x^n)$. Thus for $p \in U$, $(x^1 (p) , \dots , x^n(p))$ is a point in $\mathbb R^n$. The functions $x^1 , \dots , x^n$ are called coordinates or local coordinates on $U$. By abuse of notation, we sometimes omit the $p$. So the notation $(x^1, \dots , x^n)$ stands alternately for local coordinates on the open set $U$ and for a point in $\mathbb R^n$.
This is what I'm envisioning (correct me if I'm wrong): Let's look at some point on the manifold $p$. Then when we "feed in" this point (which is in the domain) into $\varphi$, we "spit out" a point in Euclidean space. We can consider this point to be represented as a vector that has the coordinates stored in it like so:
$$ X = \left[ \begin{array} & x^1(p) \\ x^2(p) \\ x^3(p) \\ \; \; \; \vdots \end{array} \right]$$
Using a pseudo-programming notation the first coordinate from this vector can be obtained as $X_{[1]} = x^1 (p)$. What I believe Tu is doing is referring to this entry $X_{[1]}$ as $r^1 \circ \varphi$. More succinctly, can we view $\varphi$ as being a vector-valued function that maps some point on the manifold in the domain to a vector in the image? The set-theoretic representation may look something like
$$ \varphi = \bigg\{ \{ p^\alpha, \{ p^\alpha, X^\alpha\} \}, \{ p^\beta, \{ p^\beta, X^\beta\} \} , \cdots \bigg \} $$
I hesitate to claim Tu's notation is nonstandard, but is his notation a common way to represent indexing a vector-valued function? I've read many a textbook, but I've never seen his notation used before.
If I happen to dislike Tu's notation, then am I allowed to write (for instance) the first coordinate returned by $\varphi$ as being $\varphi^{(1)} (p)$, such that $\varphi^{(1)}$ is a component of a vector-valued function?
COMMENT:
You should choose whatever notation you find easiest to use. We all have to do that because there is no standard notation. You end up translating whatever you read into your own preferred notation. So if you want to denote the $k$-th coordinate function by $\phi^k$, you should do it. I see no problem with doing so. I’m pretty sure I once insisted on doing that because I found $x^k$ to be too confusing. But it really is a commonly used convention. I eventually became comfortable with and switched to it.
Note that each coordinate can be viewed as both an input parameter for a coordinate curve and an output function as described here. The former leads to the definition of a basis of coordinate vector fields. The latter leads to a basis of coordinate 1-forms. This is also the dual basis of the basis of coordinate vector fields.