Editing Vector field (section)

==Examples==
[[File:Cessna 182 model-wingtip-vortex.jpg|thumb|250px|The flow field around an airplane is a vector field in '''R'''<sup>3</sup>, here visualized by bubbles that follow the [[Streamlines, streaklines, and pathlines|streamline]]s showing a [[wingtip vortex]].]]
[[File:Bezier curves composition ray-traced in 3D.png|thumb|Vector fields are commonly used to create patterns in [[computer graphics]]. Here: abstract composition of curves following a vector field generated with [[OpenSimplex noise]].]]
* A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length ([[Magnitude (mathematics)|magnitude]]) of the arrow will be an indication of the wind speed. A "high" on the usual [[barometric pressure]] map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
* [[Velocity]] field of a moving [[fluid]].  In this case, a [[velocity]] vector is associated to each point in the fluid.
* [[Streamlines, Streaklines and Pathlines|Streamlines, streaklines and pathlines]] are 3 types of lines that can be made from (time-dependent) vector fields. They are:
** streaklines: the line produced by particles passing through a specific fixed point over various times
** pathlines: showing the path that a given particle (of zero mass) would follow.
** streamlines (or fieldlines): the path of a particle influenced by the instantaneous field (i.e., the path of a particle if the field is held fixed).
* [[Magnetic field]]s. The fieldlines can be revealed using small [[iron]] filings.
* [[Maxwell's equations]] allow us to use a given set of initial and boundary conditions to deduce, for every point in [[Euclidean space]], a magnitude and direction for the [[force]] experienced by a charged test particle at that point; the resulting vector field is the [[electric field]].
* A [[gravitational field]] generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center with the magnitude of the vectors reducing as radial distance from the body increases.

===Gradient field in Euclidean spaces===
[[File:Irrotationalfield.svg|thumb|300px|A vector field that has circulation about a point cannot be written as the gradient of a function.]]
{{further|Gradient}}

Vector fields can be constructed out of [[scalar field]]s using the [[gradient]] operator (denoted by the [[del]]: ∇).<ref>{{cite book|author=Dawber, P.G. | title=Vectors and Vector Operators| publisher=CRC Press| isbn=978-0-85274-585-4| year=1987| page=29 |url=https://books.google.com/books?id=luBlL7oGgUIC&pg=PA29}}</ref>

A vector field ''V'' defined on an open set ''S'' is called a '''gradient field''' or a '''[[conservative field]]''' if there exists a real-valued function (a scalar field) ''f'' on ''S'' such that
<math display="block">V = \nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \frac{\partial f}{\partial x_3}, \dots ,\frac{\partial f}{\partial x_n}\right).</math>

The associated [[Flow (mathematics)|flow]] is called the '''{{visible anchor|gradient flow}}''', and is used in the method of [[gradient descent]].

The [[line integral|path integral]] along any [[closed curve]] ''γ'' (''γ''(0) = ''γ''(1)) in a conservative field is zero:
<math display="block"> \oint_\gamma V(\mathbf {x})\cdot \mathrm{d}\mathbf {x} = \oint_\gamma \nabla f(\mathbf {x}) \cdot \mathrm{d}\mathbf {x}  = f(\gamma(1)) - f(\gamma(0)).</math>

===Central field in euclidean spaces===
A {{math|''C''<sup>∞</sup>}}-vector field over {{math|'''R'''<sup>''n''</sup> \ {0}<nowiki/>}} is called a '''central field''' if
<math display="block">V(T(p)) = T(V(p)) \qquad (T \in \mathrm{O}(n, \R))</math>
where {{math|O(''n'', '''R''')}} is the [[orthogonal group]]. We say central fields are [[invariant (mathematics)|invariant]] under [[Orthogonal matrix|orthogonal transformations]] around 0.

The point 0 is called the '''center''' of the field.

Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.