Curve

Statements

Definition (@Boo75 pp. 22). Let $(t_s,t_e)$ be an interval of $\mathbb{R}$, a curve (resp. of class ${\mathcal{C}}^r$) passing through a point $a\in {\mathbb{R}}^n$ is a mapping

c:&(t_s,t_e)&\to&\mathbb{R}^n\\ &t&\mapsto&(x^1(t),\ldots,x^n(t)) \end{array}

for which there exists a $t_a\in (t_s,t_e)$ such that $c(t_a)=a$ (resp. where, for every $i\in {\mathbb{N}}_{[1,n]}$, the function $t\mapsto x^i(t)$ is of class ${\mathcal{C}}^r$).