Editing Vector field (section)

===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").