Definition:Velocity of Smooth Curve

Definition
Let $I \subseteq \R$ be an interval.

Let $\gamma: \R \to \R^n$ be a smooth curve, written in standard coordinates as:


 * $\map \gamma t = \tuple {\map {\gamma^1} t, \ldots, \map {\gamma^n} t}$.

The velocity of $\gamma$ at $t \in I$ is defined as:


 * $\map {\gamma'} t = \valueat{\map {\dot \gamma^1} t \dfrac {\partial}{\partial x^1} } {\map \gamma t} + \ldots + \valueat {\map {\dot \gamma^n} t \dfrac {\partial}{\partial x^n} } {\map \gamma t}$