Curve

Functions of the form \(f : I \rightarrow \mathbb{R}^n\) where \(I \subseteq \mathbb{R}\) are called curves. The co-domain does not necessarily have to be the \(n\) dimensional Euclidean plane.

Curves are often parametrized such that we have \(f : t \rightarrow \mathbf{x}(t)\), where \(t \in I\). As a shorthand, the curve can be denoted as \(\mathbf{x}\). Here we implicitly assume that the range of the function has the standard basis, that is, \(\mathbf{x} = [x^1, \cdots, x^m]^T = x^i \mathbf{e}_i\) where the last term uses Einstein summation convention.