Coordinate Systems

Definition: Real Vector-Valued Function

A real vector-valued function $f:D\to {\mathbb{R}}^n$ is a function from an arbitrary set $D$ to the real vector space ${\mathbb{R}}^n$.

Note: Component Functions

Every real vector-valued function $f:D\to {\mathbb{R}}^n$ can be described by $n$ real-valued functions $f_1,\dots ,f_n:\mathcal{D}\to {\mathbb{R}}^n$, where $f_k$ is responsible for the $k$-th component of the resulting vector:

$$f(x)=\left[\begin{array}{c}{f}_1(x)\\ ⋮\\ f_n(x)\end{array}\right]$$

Hence, $f_1,\dots ,f_n$ are called the component functions of $f$.