Scalar Field and Vector Field

Vector Fields and Scalar Fields

Summary

Vector fields are functions that map vectors to other vectors. Scalar fields are functions that map vectors to scalars. A scalar field is a vector field with only one output dimension.

A vector field $\vec{F}:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a function that takes as input an $n$ dimensional vector and maps it to a $m$ dimensional vector:

$$\vec{F}(x_1,\dots ,x_n)=\left[\begin{array}{c}{F}_1(x_1,\dots ,x_n)\\ ⋮\\ F_m(x_1,\dots ,x_n)\end{array}\right]$$

The $i$th output of $\vec{F}(x_1,\dots ,x_n)$ is $F_i(x_1,\dots ,x_n)$, where each $F_i:{\mathbb{R}}^n\to \mathbb{R}$ is a scalar field that depends on the $n$ inputs $x_1,\dots ,x_n$ and returns a scalar. In other words, a scalar field is a vector field whose output dimension $m=1$.