Cartesian Coordinates#
Cartesian coordinates are the simplest and most commonly used coordinates. They identify each point (vector) in the Euclidean space \(\mathbb{R}^n\) by its components. Essentially, the \(k\)-th Cartesian coordinate of \(\mathbf{p}\) is its \(k\)-th component.
Theorem: Cartesian Coordinate System
The function \(x: \mathbb{R}^n \to \mathbb{R}^n\) defined for each \(\mathbf{p} = \begin{bmatrix}p_1 & \cdots & p_n\end{bmatrix} \in \mathbb{R}^n\) as
where \(x^k = p_k\) (\(k = 1, \dotsc, n\)) is a coordinate system for \(\mathbb{R}^n\).
Proof
TODO
Notation
We denote coordinates using superscripts instead of subscripts, i.e. \(x^1, \dotsc, x^n\) instead of \(x_1, \dotsc, x_n\), in order to make it clear that we are talking about coordinates and not components. However, since the values of a point's Cartesian coordinates coincide with the values of its components, you might sometimes see the Cartesian coordinates labelled using subscripts.
Notation: Cartesian Coordinates in 2D and 3D
Cartesian coordinates in \(\mathbb{R}^2\) are usually denoted as \(x,y\) and in \(\mathbb{R}^3\) as \(x,y,z\).