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 \(\phi: \mathbb{R}^n \to \mathbb{R}^n\) 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 \(\phi\) are \(n\) functions \(\phi^1, \dotsc, \phi^n: \mathbb{R}^n \to \mathbb{R}\). These functions are known as coordinates on \(\mathbb{R}^n\). Given a specific \(\mathbf{p} \in \mathbb{R}^n\), the value \(p^i = \phi^i (\mathbf{p})\) is known as the \(i\)-th coordinate of \(\mathbf{p}\).
Notation
Coordinates are usually denotes using superscripts instead of subscripts: \(\phi^1, \dotsc, \phi^n\) and \(p^1, \dotsc, p^n\) instead of \(\phi_1, \dotsc, \phi_n\) and \(p_1, \dotsc, p_n\).
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 \(\mathbf{p} \ne \mathbf{p}'\), then at least one of \(\phi^1(\mathbf{p}), \dotsc, \phi^n(\mathbf{p})\) must be different from \(\phi^1(\mathbf{p}'), \dotsc, \phi^n(\mathbf{p}')\). Furthermore, if the Inner Product Spaces between \(\mathbf{p}\) and \(\mathbf{p}'\) is small, then the differences between \(\phi^1(\mathbf{p}), \dotsc, \phi^n(\mathbf{p})\) and \(\phi^1(\mathbf{p}'), \dotsc, \phi^n(\mathbf{p}')\), respectively, should be small.
Coordinate Tuples#
Although \(\phi\) takes in a vector \(\mathbf{p} \in \mathbb{R}^n\) and outputs the vector \(\begin{bmatrix} \phi^1(\mathbf{p}) & \cdots & \phi^n(\mathbf{p}) \end{bmatrix}^{\mathsf{T}}\), we rarely treat its output as such. It is much more common to treat its output as the \(n\)-tuple \((\phi^1 (\mathbf{p}), \dotsc, \phi^n (\mathbf{p}))\). We just define \(\phi\) 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 \(\phi^1(\mathbf{p}) + \phi^1 (\mathbf{p}')\), \(\dotsc\), \(\phi^n (\mathbf{p}) + \phi^n (\mathbf{p}')\). Whilst this might be true in certain coordinate system, it is the exception rather than the rule.