Real Vector Functions#
Definition: Real Vector-Valued Function
A real vector-valued function \(f: \mathcal{D} \to \mathbb{R}^n\) is a function from an arbitrary set \(\mathcal{D}\) to the Euclidean space \(\mathbb{R}^n\).
Note: Component Functions
Every real vector-valued function \(f: \mathcal{D} \to \mathbb{R}^n\) can be described by \(n\) real-valued functions \(f_1, \dotsc, f_n: \mathcal{D} \to \mathbb{R}\), where \(f_i(x)\) gives the \(i\)-th component of the vector \(f(x)\):
Hence, \(f_1, \dotsc, f_n\) are called the component functions of \(f\).
Definition: Vector Function
A real vector function \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) is a real vector-valued function whose domain a subset \(\mathcal{D}\) is a subset of a Euclidean space \(\mathbb{R}^m\).
Note: Component Functions
Every real vector function \(f: \mathcal{D} \to \mathbb{R}^n\) can be described by \(n\) real scalar fields \(f_1, \dotsc, f_n: \mathcal{D} \to \mathbb{R}\) such that for each vector \(\mathbb{x} \in \mathcal{D}\), the function \(f_k\) gives the the \(k\)-th component of \(f(\mathbf{x})\), which is a vector in \(\mathbb{R}^n\).
Hence, \(f_1, \dotsc, f_n\) are called the component functions of \(f\).
Notation: Multivariate Notation and Coordinate Representations
Strictly speaking, a real vector function \(f\) takes a real vector \(\mathbf{p} \in \mathcal{D} \subseteq \mathbb{R}^m\) and outputs another real vector \(f(\mathbf{p}) \in \mathbb{R}^n\). However, since real vectors live in a Euclidean space, they can be uniquely represented by coordinates.
If \((p^1, \dotsc, p^m)\) are the coordinates of \(\mathbf{p}\) with respect to some chosen coordinate system, then we can write \(f(p^1, \dotsc, p^m)\) instead of \(f(\mathbf{p})\) as long as it is which coordinate system we are working in. Unless otherwise indicated, always assume Cartesian coordinates
This is why vector functions are often called multivariate functions.