Vector field

A vector field is a vector function that assigns a vector to each point in a region.

This is contrasted with a scalar field that merely assigns a scalar value to each point.

Formulation in Set Theory

If $S$ is a subset of the $\mathbb {R} ^{n}$, then a vector field can be seen as the function ${\vec {V}}:S\rightarrow \mathbb {R} ^{n}$, which maps an n-dimensional positional vector ${\vec {x}}=(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}$ to each point ${\vec {p}}\in S$:

${\vec {V}}({\vec {p}})={\vec {x}}$