Editing Vector field (section)

===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.