Definition:Derivative of Smooth Path/Real Cartesian Space

Definition

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 $\rho_k$ with respect to $t$.

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 $\R^n$.