Coordinate Systems for Euclidean Space

Each vector $\mathbf{p}$ in the Euclidean space ${\mathbb{R}}^n$ is completely specified by its $n$ components. However, many problems are very difficult or outright impossible to solve if we are doing calculations based on vectors’ components. Hence, it is often convenient to specify each $\mathbf{p}$ in some other a way known as a coordinate system.

Definition: Coordinate System for Euclidean Space

A coordinate system $\varphi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ for the Euclidean space ${\mathbb{R}}^n$ is a continuous function which is bijective onto its image and has a continuous continuous inverse. The component functions of $\varphi$ are $n$ functions ${\varphi }^1,\dots ,{\varphi }^n:{\mathbb{R}}^n\to \mathbb{R}$. These functions are known as coordinates on ${\mathbb{R}}^n$. Given a specific $\mathbf{p}\in {\mathbb{R}}^n$, the value $p^i={\varphi }^i(\mathbf{p})$ is known as the $i$-th coordinate of $\mathbf{p}$.

NOTATION

Coordinates are usually denotes using superscripts instead of subscripts: ${\varphi }^1,\dots ,{\varphi }^n$ and $p^1,\dots ,p^n$ instead of ${\varphi }_1,\dots ,{\varphi }_n$ and $p_1,\dots ,p_n$.

INTUITION

The requirements in the definition of a coordinate system mean that each vector is identified by a unique combination of coordinates but also ensure that vector which are closed to each other have coordinates which are close to each other. Essentially, if $\mathbf{p}\ne {\mathbf{p}}^{\prime }$, then at least one of ${\varphi }^1(\mathbf{p}),\dots ,{\varphi }^n(\mathbf{p})$ must be different from ${\varphi }^1({\mathbf{p}}^{\prime }),\dots ,{\varphi }^n({\mathbf{p}}^{\prime })$. Furthermore, if the Euclidean distance between $\mathbf{p}$ and ${\mathbf{p}}^{\prime }$ is small, then the differences between ${\varphi }^1(\mathbf{p}),\dots ,{\varphi }^n(\mathbf{p})$ and ${\varphi }^1({\mathbf{p}}^{\prime }),\dots ,{\varphi }^n({\mathbf{p}}^{\prime })$, respectively, should be small.

Coordinate Tuples

Although $\varphi$ takes in a vector $\mathbf{p}\in {\mathbb{R}}^n$ and outputs the vector ${\left[\begin{array}{ccc}{\varphi }^1(\mathbf{p})& \cdots & {\varphi }^n(\mathbf{p})\end{array}\right]}^{\mathsf{T}}$, we rarely treat its output as such. It is much more common to treat its output as the $n$-tuple $({\varphi }^1(\mathbf{p}),\dots ,{\varphi }^n(\mathbf{p}))$. We just define $\varphi$ as a vector function only because it makes formulating the necessary requirements easier. This is, nevertheless, a very important point to make as treating the coordinates a vector falsely lead one to believe that the coordinates of $\mathbf{p}+\mathbf{p}$ are just ${\varphi }^1(\mathbf{p})+{\varphi }^1({\mathbf{p}}^{\prime })$, $\dots$, ${\varphi }^n(\mathbf{p})+{\varphi }^n({\mathbf{p}}^{\prime })$. Whilst this might be true in certain coordinate system, it is the exception rather than the rule.