Editing Vector field (section)

{{short description|Assignment of a vector to each point in a subset of Euclidean space}}
[[File:VectorField.svg|right|thumb|250px|A portion of the vector field (sin&nbsp;''y'',&nbsp;sin&nbsp;''x'')]]

In [[vector calculus]] and [[physics]], a '''vector field''' is an assignment of a [[vector (mathematics and physics)|vector]] to each point in a [[Space (mathematics)|space]], most commonly [[Euclidean space]] <math>\mathbb{R}^n</math>.<ref name="Galbis-2012-p12" />  A vector field on a [[Plane (geometry)|plane]] can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout [[three dimensional space]], such as the [[wind]], or the strength and direction of some [[force]], such as the [[magnetic field|magnetic]] or [[gravity|gravitational]] force, as it changes from one point to another point.

The elements of [[differential and integral calculus]] extend naturally to vector fields.  When a vector field represents [[force]], the [[line integral]] of a vector field represents the [[Work (physics)|work]] done by a force moving along a path, and under this interpretation [[conservation of energy]] is exhibited as a special case of the [[fundamental theorem of calculus]].  Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the [[divergence]] (which represents the rate of change of [[volume]] of a flow) and [[curl (mathematics)|curl]] (which represents the rotation of a flow).

A vector field is a special case of a ''[[vector-valued function]]'', whose domain's dimension has no relation to the dimension of its range; for example, the [[position vector]] of a [[space curve]] is defined only for smaller subset of the ambient space.
Likewise, n [[Coordinate system|coordinates]], a vector field on a domain in ''n''-dimensional Euclidean space <math>\mathbb{R}^n</math> can be represented as a vector-valued function that associates an ''n''-tuple of real numbers to each point of the domain.  This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law (''[[covariance and contravariance of vectors]]'') in passing from one coordinate system to the other.

Vector fields are often discussed on [[open set|open subsets]] of Euclidean space, but also make sense on other subsets such as [[Surface (topology)|surface]]s, where they associate an arrow tangent to the surface at each point (a [[Differential geometry of curves|tangent vector]]).
More generally, vector fields are defined on [[differentiable manifold]]s, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales.  In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a [[Section (fiber bundle)|section]] of the [[tangent bundle]] to the manifold).  Vector fields are one kind of [[tensor field]].