\begin{definition}[Definition:Directed Smooth Curve]
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 $\rho$ {{iff}}:
: there exists a bijective differentiable strictly increasing real function:
:: $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$
: such that $\sigma = \rho \circ \phi$.
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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 $\rho$.
\end{definition}