\begin{definition}[Definition:Derivative of Smooth Path/Real Cartesian Space]
Let $\R^n$ be a real cartesian space of $n$ dimensions.
Let $\closedint a b$ be a closed real interval.
Let $\rho: \closedint a b \to \R^n$ be a '''smooth path in $\R^n$'''.
For each $k \in \set {1, 2, \ldots, n}$, define the real function $\rho_k: \closedint a b \to \R$ by:
:$\forall t \in \closedint a b: \map {\rho_k} t = \map {\pr_k} {\map \rho t}$
where $\pr_k$ denotes the $k$th projection from the image $\Img \rho$ of $\rho$ to $\R$.
It follows from the definition of a  smooth path that $\rho_k$ is continuously differentiable for all $k$.
Let $\map {\rho_k'} t$ denote the derivative of $\rho_k$ {{WRT|Differentiation}} $t$.
The '''derivative of $\rho$''' is the continuous vector-valued function $\rho': \closedint a b \to \R^n$ defined by:
:$\ds \forall t \in \closedint a b: \map {\rho'} t = \sum_{k \mathop = 1}^n \map {\rho_k'} t \mathbf e_k$
where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ denotes the standard ordered basis of $\R^n$.
Category:Definitions/Vector Analysis
Category:Definitions/Vector Calculus
\end{definition}