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/* More formal treatment */Category
A '''vector field''' is an assignment of a [[vector]] ''function'' that assigns a vector to each point in a some regionof 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]].
In analogy with a vector field, a [[scalar field]] is an assignment of a plain number ("scalar") to each point. There are tensor fields as well, assigning a [[tensor]] to every point in space. == Formulation in Set Theory More formal treatment == A vector field is a ''function'' that maps the region of space into the [[vector space]].
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>:
<center><math>\vec{V}(\vec{p}) = \vec{x}</math></center>
[[Category:vector analysisMathematics]][[Category:calculusAlgebra]][[Category:mathematicsVector Analysis]][[Category:Calculus]][[Category:Physics]]
