Definition:Smooth Curve/3-Dimensional Real Vector Space
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Definition
Let $\R^3$ be the $3$-dimensional real vector space.
Let $I$ be an bounded or unbounded open interval.
A smooth curve in $\R^3$ is a mapping $\alpha : I \to \R^3$ defined as:
- $\map \alpha t := \tuple {\map x t, \map y t, \map z t}$
where $\map x t, \map y t, \map z t$ are smooth real functions.
Also known as
In the literature, the $\alpha$ is also called a parameterized (or parametrized) differentiable curve.
Sources
- 2016: Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (2nd ed.): $1$-$2$: Parametrized Curves