Editing Vector field (section)

==Definition==

===Vector fields on subsets of Euclidean space===
{{multiple image
| footer    = Two representations of the same vector field: {{nowrap|1='''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 {{math|''S''}} of {{math|'''R'''<sup>''n''</sup>}}, a '''vector field''' is represented by a [[vector-valued function]] {{math|''V'': ''S'' → '''R'''<sup>''n''</sup>}} in standard [[Cartesian coordinates]] {{math|(''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>)}}. If each component of {{math|''V''}} is continuous, then {{math|''V''}} is a continuous vector field. It is common to focus on '''smooth''' vector fields, meaning that each component is a [[smooth function]] (differentiable any number of times). 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|author1=Galbis, Antonio |author2=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>

One standard notation is to write <math>\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_n}</math> for the unit vectors in the coordinate directions. In these terms, every smooth vector field <math>V</math> on an open subset <math>S</math> of <math>{\mathbf R}^n</math> can be written as
:<math> \sum_{i=1}^n V_i(x_1,\ldots,x_n)\frac{\partial}{\partial x_i}</math>
for some smooth functions <math>V_1,\ldots,V_n</math> on <math>S</math>.<ref name="Tu-2010-p149" /> The reason for this notation is that a vector field determines a [[linear map]] from the space of smooth functions to itself, <math>V\colon C^{\infty}(S)\to C^{\infty}(S)</math>, given by differentiating in the direction of the vector field.

'''Example''': The vector field <math>-x_2\frac{\partial}{\partial x_1}+x_1\frac{\partial}{\partial x_2}</math> describes a counterclockwise rotation around the origin in <math>\mathbf{R}^2</math>. To show that the function <math>x_1^2+x_2^2</math> is rotationally invariant, compute:
:<math>\bigg(-x_2\frac{\partial}{\partial x_1}+x_1\frac{\partial}{\partial x_2}\bigg)(x_1^2+x_2^2) = -x_2(2x_1)+x_1(2x_2) = 0.</math>

Given vector fields {{math|''V''}}, {{math|''W''}} defined on {{math|''S''}} and a smooth function {{mvar|f}} defined on {{math|''S''}}, the operations of scalar multiplication and vector addition,
<math display="block"> (fV)(p) := f(p)V(p)</math>
<math display="block"> (V+W)(p) := V(p) + W(p),</math>
make the smooth vector fields into a [[Module (mathematics)|module]] over the [[Ring (mathematics)|ring]] of smooth functions, where multiplication of functions is defined pointwise.

===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 {{math|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} is a choice of Cartesian coordinates, in terms of which the components of the vector {{mvar|V}} are
<math display="block">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 display="block">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 field E.png|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 name="Tu-2010-p149">{{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|chapter-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) \to 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>{{cite web |title=An Introduction to Differential Geometry |first=Eugene |last=Lerman |date=August 19, 2011 |url=https://faculty.math.illinois.edu/~lerman/518/f11/8-19-11.pdf#page=18 |at=Definition 3.23 }}</ref>

If the manifold <math>M</math>  is smooth or [[analytic function|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 display="inline"> \mathfrak{X} (M)</math> (a [[fraktur (typeface sub-classification)|fraktur]] "X").