What's the point in Coordinate Functions? A long time ago I've asked here about what O'Neill defines in his "Elementary Differential Geometry" book as "Natural Coordinate Functions". In the time, I've understood that it was a notational convenience and all of that. But now I've started to thing again about it, and I just want to make sure I've understood what's happening.
O'Neill in his book, instead of presenting functions as: "let $f: \Bbb R^2 \to \Bbb R$ be given by the relation $f(x,y)=x^2+y^2$" where $x,y \in \Bbb R$ he presents things like "let $f: \Bbb R^2 \to \Bbb R$ be given by the folllowing $f=x^2+y^2$" where he understands $x$ and $y$ to be the functions such that $x(a,b)=a$ and $y(a,b)=b$.
Now, in $\Bbb R^n$ it doesn't really seem necessary to work as that, but now it comes the question: is this the way we usually should setup function in abstract manifolds? Indeed, let $M$ be a smooth manifold of dimension $n$, and let $(x,U)$ be a chart for $M$. We have then $x: U \subset M \to \Bbb R^n$, if $I : \Bbb R^n \to \Bbb R$ is the identity function we can then extend this idea of O'neill's book and define $x^i = I^i \circ x$ to be the $i$-th coordinate function. Now, we can set up every function $f : U \to \Bbb R$ in terms of these functions, so that we can write for instance (if $n = 2$) $f = (x^1)^2 + (x^2)^2$ and so we have a very natural way to write things in $U$ in terms of "the coordinates" $(x^1,\cdots,x^n)$. So that instead of really writing $f(p)$ for $p \in M$ we just write $f$ as a combination of the coordinate functions (using addition, multiplication by scalar, product, composition and so on)?
Is that right? Is really like this that this thing of "coordinate functions" really work?
Thanks very much in advance.
 A: Well this might not be what you expected, but an example of using this thinking could be what I am wrestling with now.  I am researching a machine design that mills material (wood & plastic) in polar coordinate space r,theta but uses GCode programming for Cartesian space(x,y,z  and I,J,K displacements).
A traditional milling-machine has an x,y table and a cutting head or drill that moves up & down in the z direction. Mechanical tolerances are tight with this method.
Imagine a pillar drill ... I want to pivot the drill-head in an arc around the main support pillar and rotate the drill table (x,y plane) ie pivot it around the axis parallel to the support pillar. Think of a gramophone turntable/platter and the needle-arm pivoting over the platter. One should be able to reach every part of the platter surface by judicious rotation of the turntable and needle-arm.
I Need to solve how to map these functions and then write some code.
Productive workflow will be to take an engineering drawing as a .dxf file and convert to CNC milling-machine Gcode ... this is standard fare ... but now we need the transform to polar cords. for the stepper-motors that will rotate the 'turntable' and 'needle-arm' and move the drill-head up and down (z axis).
Hopefully you see some point in all this ?
