Definition:Directed Smooth Curve

Definition
Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$.

The directed smooth curve with parameterization $\rho$ is defined as an equivalence class of smooth paths as follows:

A smooth path $\sigma: \left[{a \,.\,.\, b}\right] \to \R^n$ belongs to the equivalence class of $\rho$ :
 * there exists a bijective differentiable strictly increasing real function:
 * $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$
 * such that $\sigma = \rho \circ \phi$.

It follows from Directed Smooth Curve Relation is Equivalence and Fundamental Theorem on Equivalence Relations that this does in fact define an equivalence class.

If a directed smooth curve is only defined by a smooth path $\rho$, then it is often denoted with the same symbol $\rho$.

Complex Plane
The definition carries over to the complex plane, in which context it is usually applied:

Also known as
A directed smooth curve is called an oriented smooth curve, a smooth curve with orientation or simply a curve in many texts.