View source for Vector field

{{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 (geometry)|vector]] to each point in a subset of [[Space (mathematics)|space]].<ref name="Galbis-2012-p12" />  A vector field in the plane (for instance), can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane.  Vector  fields are often used to model, for example, the speed and direction of a moving fluid throughout space, 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).

In coordinates, a vector field on a domain in ''n''-dimensional [[Euclidean space]] 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 [[covariance and contravariance of vectors|transformation law]] 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]].

==Definition==

===Vector fields on subsets of Euclidean space===
{{multiple image
| footer    = Two representations of the same vector field: '''v'''(''x'', ''y'') = −'''r'''. The arrows depict the field at discrete points, however, the field exists everywhere.
| width     = 140

| image1 = Radial_vector_field_sparse.svg
| alt1      = Sparse vector field representation

| image2 = Radial_vector_field_dense.svg
| alt2      = Dense vector field representation.
}}

Given a subset ''S'' in '''R'''<sup>''n''</sup>, a '''vector field''' is represented by a [[vector-valued function]] ''V'': ''S'' → '''R'''<sup>''n''</sup> in standard Cartesian coordinates (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>).  If each component of ''V'' is continuous, then ''V'' is a continuous vector field, and more generally ''V'' is a ''C<sup>k</sup>'' vector field if each component of ''V'' is ''k'' times [[continuously differentiable]].

A vector field can be visualized as assigning a vector to individual points within an ''n''-dimensional space.<ref name="Galbis-2012-p12">{{cite book|authors=Galbis, Antonio & Maestre, Manuel|title=Vector Analysis Versus Vector Calculus|publisher=Springer|year=2012|isbn=978-1-4614-2199-3|page=12|url=https://books.google.com/books?id=tdF8uTn2cnMC&pg=PA12}}</ref>

Given two ''C<sup>k</sup>''-vector fields ''V'', ''W'' defined on ''S'' and a real valued ''C<sup>k</sup>''-function ''f'' defined on ''S'', the two operations scalar multiplication and vector addition

:<math> (fV)(p) := f(p)V(p)\,</math>
:<math> (V+W)(p) := V(p) + W(p)\,</math>

define the [[Module (mathematics)|module]] of ''C<sup>k</sup>''-vector fields over the [[Ring (mathematics)|ring]] of C<sup>''k''</sup>-functions where the multiplication of the functions is defined pointwise (therefore, it is commutative with the multiplicative identity being ''f<sub>id</sub>(p) := 1'').

===Coordinate transformation law===
In physics, a [[Euclidean vector|vector]] is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system.  The [[Euclidean vector#Vectors, pseudovectors, and transformations|transformation properties of vectors]] distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a [[covector]].

Thus, suppose that (''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) is a choice of Cartesian coordinates, in terms of which the components of the vector ''V'' are

:<math>V_x = (V_{1,x},\dots,V_{n,x})</math>

and suppose that (''y''<sub>1</sub>,...,''y''<sub>''n''</sub>) are ''n'' functions of the ''x''<sub>''i''</sub> defining a different coordinate system.  Then the components of the vector ''V'' in the new coordinates are required to satisfy the transformation law
{{NumBlk|:|<math>V_{i,y} = \sum_{j=1}^n \frac{\partial y_i}{\partial x_j} V_{j,x}.</math>|{{EquationRef|1}}}}

Such a transformation law is called [[covariance and contravariance of vectors|contravariant]].  A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of ''n'' functions in each coordinate system subject to the transformation law ({{EquationNote|1}}) relating the different coordinate systems.

Vector fields are thus contrasted with [[scalar field]]s, which associate a number or ''scalar'' to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.

===Vector fields on manifolds===
[[File:Vector sphere.svg|right|200px|thumb|A vector field on a [[sphere]]]]
Given a [[differentiable manifold]] <math>M</math>, a '''vector field''' on <math>M</math> is an assignment of a [[Tangent space|tangent vector]] to each point in <math>M</math>.<ref>{{cite book|author=Tu, Loring W.|chapter=Vector fields|title=An Introduction to Manifolds|publisher=Springer|year=2010|isbn=978-1-4419-7399-3|page=149|url=https://books.google.com/books?id=PZ8Pvk7b6bUC&pg=PA149}}</ref> More precisely, a vector field <math>F</math> is a [[Map (mathematics)|mapping]] from <math>M</math> into the [[tangent bundle]] <math>TM</math> so that <math> p\circ F </math> is the identity mapping
where <math>p</math> denotes the projection from <math>TM</math> to <math>M</math>. In other words, a vector field is a [[section (fiber bundle)|section]] of the [[tangent bundle]].

An alternative definition: A smooth vector field <math>X</math> on a manifold <math>M</math> is a linear map <math>X: C^\infty(M) \rightarrow C^\infty(M)</math> such that <math>X</math> is a [[Derivation_(differential_algebra)|derivation]]:
<math>X(fg)=fX(g)+X(f)g</math> for all <math>f,g \in C^\infty(M)</math> <ref>http://www.math.uiuc.edu/~lerman/518/f11/8-19-11.pdf</ref>.

If the manifold <math>M</math>  is smooth or analytic—that is, the change of  coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold <math>M</math> is often denoted by <math>\Gamma (TM)</math> or <math>C^\infty (M,TM)</math> (especially when thinking of vector fields as [[section (fiber bundle)|section]]s); the collection of all smooth vector fields is also denoted by <math>\scriptstyle \mathfrak{X} (M)</math> (a [[fraktur (typeface sub-classification)|fraktur]] "X").

==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]].]]
* 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  &mdash; the line produced by particles passing through a specific fixed point over various times
::pathlines &mdash; showing the path that a given particle (of zero mass) would follow.
::streamlines (or fieldlines)&mdash; 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 [[electromagnetic 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.]]
{{details|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>V = \nabla f = \bigg(\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}\bigg).</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> \oint_\gamma V(\boldsymbol {x})\cdot \mathrm{d}\boldsymbol {x}  = \oint_\gamma \nabla f(\boldsymbol {x})\cdot \mathrm{d}\boldsymbol {x}  = f(\gamma(1)) - f(\gamma(0)).</math>

===Central field in euclidean spaces===
A ''C''<sup>∞</sup>-vector field over '''R'''<sup>''n''</sup> \ {0} is called a '''central field''' if

:<math>V(T(p)) = T(V(p)) \qquad (T \in \mathrm{O}(n, \mathbf{R}))</math>

where 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.

==Operations on vector fields==

===Line integral===
{{Main|Line integral}}
A common technique in physics is to integrate a vector field along a [[differential geometry of curves|curve]], also called determining its [[line integral]]. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on the particle, when it travels along this path. Intuitively, it is the sum of the scalar products of the force vector and the small tangent vector in each point along the curve.

The line integral is constructed analogously to the [[Riemann integral]] and it exists if the curve is rectifiable (has finite length) and the vector field is continuous.

Given a vector field ''V'' and a curve γ, [[parametric equation|parametrized]] by ''t'' in [''a'', ''b''] (where ''a'' and ''b'' are [[real number]]s), the line integral is defined as

:<math>\int_\gamma  V(\boldsymbol {x})\cdot \mathrm{d}\boldsymbol {x}  = \int_a^b  V(\gamma(t))\cdot\dot \gamma(t)\; \mathrm{d}t.</math>

===Divergence===
{{Main|Divergence}}
The [[divergence]] of a vector field on Euclidean space is a function (or scalar field).  In three-dimensions, the divergence is defined by

:<math>\operatorname{div} \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z},</math>

with the obvious generalization to arbitrary dimensions.  The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the [[divergence theorem]].

The divergence can also be defined on a [[Riemannian manifold]], that is, a manifold with a [[Riemannian metric]] that measures the length of vectors.

===Curl in three-dimensions===
{{Main|Curl  (mathematics)}}
The [[Curl (mathematics)|curl]] is an operation which takes a vector field and produces another vector field. The curl is defined only in three-dimensions, but some properties of the curl can be captured in higher dimensions with the [[exterior derivative]].  In three-dimensions, it is defined by

:<math>\operatorname{curl}\,\mathbf{F} = \nabla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y}- \frac{\partial F_2}{\partial z}\right)\mathbf{e}_1 - \left(\frac{\partial F_3}{\partial x}- \frac{\partial F_1}{\partial z}\right)\mathbf{e}_2 + \left(\frac{\partial F_2}{\partial x}- \frac{\partial F_1}{\partial y}\right)\mathbf{e}_3.</math>

The curl measures the density of the [[angular momentum]] of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis.  This intuitive description is made precise by [[Stokes' theorem]].

===Index of a vector field===
The index of a vector field is an integer that helps to describe the behaviour of a vector field around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value -1 at a saddle singularity but +1 at a source or sink singularity.

Let the dimension of the manifold on which the vector field is defined be '''n'''. Take a small sphere S around the zero so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimensions ''n''&nbsp;−&nbsp;1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere S<sup>n-1</sup>. This defines a continuous map from S to S<sup>n-1</sup>. The index of the vector field at the point is the [[Degree of a continuous mapping#Differential topology|degree]] of this map. It can be shown that this integer does not depend on the choice of S, and therefore depends only on the vector field itself.

'''The index of the vector field''' as a whole is defined when it has just a finite number of zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes.

The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). it is equal to +1 around a source, and more generally equal to (−1)<sup>k</sup> around a saddle that has k contracting dimensions and n-k expanding dimensions. For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the [[hairy ball theorem]], which states that if a vector in R<sup>3</sup> is assigned to each point of the unit sphere S<sup>2</sup> in a continuous manner, then it is impossible to "comb the hairs flat", i.e., to choose the vectors in a continuous way such that they are all non-zero and tangent to S<sup>2</sup>.

For a vector field on a compact manifold with a finite number of zeroes, the [[Poincaré-Hopf theorem]] states that the index of the vector field is equal to the [[Euler characteristic]] of the manifold.

==Physical intuition==
[[File:Magnet0873.png|thumb|[[Magnetism|Magnetic]] field lines of an iron bar ([[magnetic dipole]])]]

[[Michael Faraday]], in his concept of ''[[lines of force]],'' emphasized that the field ''itself'' should be an object of study, which it has become throughout physics in the form of [[field theory (physics)|field theory]].

In addition to the magnetic field, other phenomena that were modeled by Faraday include the electrical field and [[light field]].

==Flow curves==
{{Main|Integral curve}}
Consider the flow of a fluid through a region of space.  At any given time, any point of the fluid has a particular velocity associated with it; thus there is a vector field associated to any flow.  The converse is also true: it is possible to associate a flow to a vector field having that vector field as its velocity.

Given a vector field ''V'' defined on ''S'', one defines curves γ(''t'') on ''S'' such that for each ''t'' in an interval ''I''

:<math>\gamma'(t) = V(\gamma(t))\,.</math>

By the [[Picard–Lindelöf theorem]], if ''V'' is [[Lipschitz continuity|Lipschitz continuous]] there is a ''unique'' ''C''<sup>1</sup>-curve γ<sub>''x''</sub> for each point ''x'' in ''S'' so that, for some ε > 0,

:<math>\gamma_x(0) = x\,</math>
:<math>\gamma'_x(t) = V(\gamma_x(t)) \qquad \forall t \in (-\varepsilon, +\varepsilon) \subset \mathbf{R}.</math>

The curves γ<sub>''x''</sub> are called '''integral curves''' or '''trajectories''' (or less commonly, flow lines) of the vector field ''V'' and partition ''S'' into [[equivalence class]]es. It is not always possible to extend the interval (−ε, +ε) to the whole [[real number line]]. The flow may for example reach the edge of ''S'' in a finite time.
 <!--Integrating the vector field along any flow curve &gamma; yields

:<math>\int_\gamma \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle = \int_a^b \langle \mathbf{F}( \mathbf{\gamma}(t) ), \mathbf{\gamma}'(t) \rangle dt = \int_a^b dt = \mbox{constant}. </math>
-->
In two or three dimensions one can visualize the vector field as giving rise to a [[Flow (mathematics)|flow]] on ''S''. If we drop a particle into this flow at a point ''p'' it will move along the curve γ<sub>''p''</sub> in the flow depending on the initial point ''p''. If ''p'' is a stationary point of ''V'' (i.e., the vector field is equal to the zero vector at the point ''p''), then the particle will remain at ''p''.

Typical applications are [[Streamlines, streaklines, and pathlines|pathline]] in [[fluid flow|fluid]], [[geodesic flow]], and [[one-parameter subgroup]]s and the [[exponential map (Lie theory)|exponential map]] in [[Lie group]]s.

===Complete vector fields===
By definition, a vector field is called '''complete''' if every one of its flow curves exist for all time.<ref>{{cite book|author=Sharpe, R.|title=Differential geometry|publisher=Springer-Verlag|year=1997|isbn=0-387-94732-9}}</ref>  In particular, [[compact support|compactly supported]] vector fields on a manifold are complete.  If  <math>X</math> is a complete vector field on <math>M</math>, then the [[one-parameter group]] of [[diffeomorphism]]s generated by the flow along <math>X</math> exists for all time. On a compact manifold without boundary, every smooth vector field is complete.  An example of an '''incomplete''' vector field <math>V</math> on the real line <math>\mathbb R</math> is given by <math>V(x) = x^2</math>.  For, the differential equation <math>\frac{dx}{dt} = x^2</math>, with initial condition <math>x(0) = x_0 </math>, has as its unique solution <math>x(t) = \frac{x_0}{1 - t x_0}</math> if  <math>x_0 \neq 0</math> (and <math>x(t) = 0</math> for all <math>t \in \mathbb R</math> if  <math>x_0 = 0</math>).  Hence for <math>x_0 \neq 0</math>, <math>x(t)</math> is undefined at <math>t = \frac{1}{x_0}</math> so cannot be defined for all values of <math>t</math>.

==f-relatedness==
Given a [[smooth function]] between manifolds, ''f'': ''M'' → ''N'', the [[derivative]] is an induced map on [[tangent bundle]]s, ''f''<sub>*</sub>: ''TM'' → ''TN''. Given vector fields ''V'': ''M'' → ''TM'' and ''W'': ''N'' → ''TN'', we say that ''W'' is ''f''-related to ''V'' if the equation ''W'' ∘ ''f'' = ''f''<sub>*</sub> ∘ ''V'' holds.

If ''V''<sub>i</sub> is ''f''-related to ''W''<sub>i</sub>, ''i'' = 1, 2, then the [[Lie bracket]] [''V''<sub>1</sub>, ''V''<sub>2</sub>] is ''f''-related to [''W''<sub>1</sub>, ''W''<sub>2</sub>].

==Generalizations==
Replacing vectors by [[p-vector|''p''-vectors]] (''p''th exterior power of vectors) yields ''p''-vector fields; taking the [[dual space]] and exterior powers yields [[differential form|differential ''k''-forms]], and combining these yields general [[tensor field]]s.

Algebraically, vector fields can be characterized as [[Derivation (abstract algebra)|derivations]] of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of [[differential calculus over commutative algebras]].

==See also==
{{Portal|Mathematics}}
{{div col|colwidth=22em}}
* [[Eisenbud–Levine–Khimshiashvili signature formula]]
* [[Field line]]
* [[Field strength]]
* [[Balanced flow#Gradient flow|''Gradient flow'' and ''balanced flow'']] in ''[[atmospheric dynamics]]''
* [[Lie derivative]]
* [[Scalar field]]
* [[Time-dependent vector field]]
* [[Vector fields in cylindrical and spherical coordinates]]
* [[Tensor fields]]
{{div col end}}

==References==
{{refimprove|date=April 2012}}
{{reflist}}

==Bibliography==
*{{cite book | last1 = Hubbard | first1 = J. H. | authorlink1 = John H. Hubbard | last2 = Hubbard | first2 = B. B. | title = Vector calculus, linear algebra, and differential forms. A unified approach | origyear = | url = | edition = | year = 1999 | publisher = Prentice Hall | location = Upper Saddle River, NJ | isbn = 0-13-657446-7 | pages = }} 
*{{cite book | last =Warner | first = Frank | title = Foundations of differentiable manifolds and Lie groups | origyear = 1971 | edition = | year = 1983 | publisher =  Springer-Verlag | location = New York-Berlin | isbn = 0-387-90894-3 | pages = }}
*{{cite book | last = Boothby | first = William | title = An introduction to differentiable manifolds and Riemannian geometry | url = https://archive.org/details/introductiontodi0000boot | url-access = registration | edition = second  | series = Pure and Applied Mathematics, volume 120 | publisher = Academic Press | location = Orlando, FL | year = 1986 | isbn = 0-12-116053-X }}

==External links==
{{Commons category|Vector fields}}
* [https://anvaka.github.io/fieldplay/ Online Vector Field Editor]
* {{springer|title=Vector field|id=p/v096420}}
* [http://mathworld.wolfram.com/VectorField.html Vector field] &mdash; [[Mathworld]]
* [http://planetmath.org/encyclopedia/VectorField.html Vector field] &mdash; [[PlanetMath]]
* [http://www.amasci.com/electrom/statbotl.html 3D Magnetic field viewer]
* [http://www-solar.mcs.st-and.ac.uk/~alan/MT3601/Fundamentals/node2.html Vector fields and field lines]
* [http://www.vias.org/simulations/simusoft_vectorfields.html Vector field simulation] An interactive application to show the effects of vector fields

{{DEFAULTSORT:Vector Field}}
[[Category:Differential topology]]
[[Category:Vector calculus]]