Real Vector Function

Definition: Vector Function

A vector function $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ is a real vector-valued function whose domain $\mathcal{D}$ is a subset of a real vector space ${\mathbb{R}}^m$.

Notation: Multivariate Notation and Coordinate Representations

Strictly speaking, a real vector function $f$ takes a vector $\mathbf{p}\in \mathcal{D}\subseteq {\mathbb{R}}^m$ and outputs another vector $f(\mathbf{p})\in {\mathbb{R}}^n$. However, since vectors live as a points in Euclidean space, they can be uniquely represented by coordinates.

Although the action of $f$ is independent of the choice of coordinate systems for ${\mathbb{R}}^m$ and ${\mathbb{R}}^n$ ($\mathbf{p}$ and $f(\mathbf{p})$ are the same vectors, regardless of how we choose to represent them), many definitions involving $f$, such as differentiability and integrability, do depend on the choice of a coordinate system for the input space ${\mathbb{R}}^m$. Indeed, it would be more appropriate to formulate such definitions about the coordinate representations of $f$ in the chosen coordinate systems. However, this formality gets very cumbersome very quickly.

Hence, if the coordinates of $\mathbf{p}$ in a given coordinate system $\varphi$ are $p^1,\dots ,p^n$, then instead of writing ${}_{({\mathbb{R}}^m,\varphi )}f(p^1,\dots ,p^m)$ for the coordinate representation of $f$, we can just write $f(p^1,\dots ,p^m)$, as long as it is clear which coordinate system we are using. If there is no particular coordinate system specified, then we usually assume that those are Cartesian coordinates.

This is why vector functions are often called multivariate functions.

Note: Component Functions

The component functions of $f$ are real scalar fields.