Definition:Velocity of Smooth Curve
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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:
- $\map {\gamma'} t = \valueat{\map {\dot \gamma^1} t \dfrac {\partial}{\partial x^1} } {\map \gamma t} + \ldots + \valueat {\map {\dot \gamma^n} t \dfrac {\partial}{\partial x^n} } {\map \gamma t}$
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. The Problem of Differentiating Vector Fields