Changes

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

A '''vector field''' is an assignment of a [[vector]] to each point in some region of space.  A simple example of a vector field is the wind velocity (a vector) at each point on the Earth's surface.  Vector fields are extremely important in science and engineering—some examples are the electric and magnetic fields of [[Maxwell's Equations|electrodynamics]], and the gravitational field of [[gravitation|Newtonian gravity]].

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

In addition to the usual operations of vectors (addition, scalar multiplication, [[dot product]], and [[cross product]]), vector fields have some differentiation operations that compare the values of the field at different points.  The common operations are the [[divergence]] and the [[curl]].

== Formulation in Set Theory ==

 

== More formal treatment ==

 

If <math>S</math> is a subset of the <math>\mathbb{R}^n</math>, then a vector field can be seen as the function <math>\vec{V}: S \rightarrow \mathbb{R}^n</math>, which maps an n-dimensional positional vector <math>\vec{x}=(x_1, x_2, \dots, x_n) \in \mathbb{R}^n</math> to each point <math>\vec{p} \in S</math>:

If <math>S</math> is a subset of the <math>\mathbb{R}^n</math>, then a vector field can be seen as the function <math>\vec{V}: S \rightarrow \mathbb{R}^n</math>, which maps an n-dimensional positional vector <math>\vec{x}=(x_1, x_2, \dots, x_n) \in \mathbb{R}^n</math> to each point <math>\vec{p} \in S</math>: