Notation of a function in coordinates

Question

I have a question about some notation which puzzles me a lot. Consider a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$. Then people often write or say that if we choose coordinates $x=(x^1,…,x^n)$ in $\mathbb{R}^n$ and $u=(u^1,…,u^m)$ in $\mathbb{R}^m$ we can think of $f$ as a collection of $m$ functions as follows: $$u^1=u^1(x^1,…,x^n),…,u^m=u^m(x^1,…,x^n).$$

My question is about the last line above. I'm sure it is just a matter of notation, but anyhow I want to grasp the logic behind this kind of notation since it appears frequently and it is not very convenient if one has to think about the notation more often than should be needed. Would someone be so kind to explain it to me?

Just to give you a taste of the questions which I have in mind:

First of all, it the functions $u^i$ are by definition coordinates (e.g. cartesian coordinates, polar coordinates etc.) which should be independent of any function $f$. At the same time it is said, that they represent the function $f$ whereas the function $f$ does not even appear in the last line above! Furthermore it looks to me as if the coordinates $(x^1,…,x^n)$ are plugged in into the functions $u^i$. But if the functions $u^i$ are coordinates of $\mathbb{R}^m$ they should depend on $m$, though not on $n$ arguments.

Any help will be very appreciated. Best wishes

Answer 1

In the beginning there just were ${\mathbb R}^n$ with coordinates $(x_1,\ldots, x_n)$ and ${\mathbb R}^m$ with coordinates $(u_1,\ldots, u_m)$. The $x_k$ as well as the $u_i$ were independent real variables encoding the position of points ${\bf x}\in{\mathbb R}^n$, resp. ${\bf u}\in{\mathbb R}^m$ living freely in the wild.

Then came a mathematician considering a certain map ${\bf f}:\>{\mathbb R}^n\to{\mathbb R}^m$ defined in so many words: by a geometric construction, as time-one-map of a dynamic system, what have you. This map moves individual points ${\bf x}\in{\mathbb R}^n$ to points ${\bf u}:={\bf f}({\bf x})\in{\mathbb R}^m$. In order to do computations with this ${\bf f}$ we (sometimes) have to use its expression in terms of coordinates. In this way ${\bf f}$ appears as a tuple map $${\bf f}:\quad(x_1,x_2,\ldots,x_n)\mapsto\left\{\eqalign{u_1&:=f_1(x_1,\ldots,x_n)\cr &\vdots\cr u_m&:=f_m(x_1,\ldots,x_n)\cr}\right.\ .$$ It is at this point that a handy abuse of notation is very common: In certain cases we are not so much interested in the map ${\bf f}$ per se, but in the object produced by ${\bf f}$ in ${\bf u}$-space: a curve $\gamma$, or a surface $S$. Therefore we just write $u_i$ for $f_i({\bf x})$ and then $\dot{\bf u}$ when we mean the tangent vector of $\gamma$, etc. This is only allowed if only one ${\bf f}$ is at stake. If we want, e.g., to argue about the angle of intersection between two different curves we have to return to the full notation involving ${\bf f}$.

Answer 2

You should look at this as follows: let $f:X \rightarrow \mathbb{R}^m$ be some function and $u^i:\mathbb{R}^m \rightarrow \mathbb{R}$ the $i$-th coordinate function (i.e. for $y=(y^1, \ldots , y^m)^t$ you have $u^i(y) = y^i$). Note this is nothing but a real valued function on $\mathbb{R}^m $.

Then what you denote as $u^i(x)$ is simply $u^i\circ f(x)$.

Answer 3

$f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ implies that $f:x=(x^1,\ldots,x^n)\mapsto f(x)=u=(u^1,\ldots,u^m)$. For each $x$, $f(x)$ inhabits $m$-dimensional space, so by definition it has $m$ coordinates; these are represented by $u^1,\ldots,u^m$. They could as easily be called $f^1(x),\ldots,f^m(x)$.

Answer 4

I think that an example is necessary. Let $f : \Bbb{R}^2 \to \Bbb{R}^3$ be defined as $$f(x,y) = (x+y, y^7, \cos(x+2y)-4)$$ then $f$ is represented in coordinates by the three functions $u^1, u^2, u^3: \Bbb{R}^2 \to \Bbb{R}$ $$u^1(x,y)= x+y$$ $$u^2(x,y)= y^7$$ $$u^3(x,y)= \cos(x+2y)-4$$