Class (Map)

A Map $u:\Bbb{R}^n\to\Bbb{R}^n$ from a Domain $G$ is called a map of class $C^r$ if each component of

$$u(x)=(u_1(x_1, \ldots, x_n), \ldots, u_m(x_1, \ldots, x_n))$$

is of class $C^r$ ( $0\leq r\leq\infty$ or $r=\omega$) in $G$, where $C^d$ denotes a continuous function which is differentiable $d$ times.