Definition:Smooth Curve/3-Dimensional Real Vector Space

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.