Editing Vector field (section)

==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 <math>V</math> defined on <math>S</math>, one defines curves <math>\gamma(t)</math> on <math>S</math> such that for each <math>t</math> in an interval <math>I</math>,
<math display="block">\gamma'(t) = V(\gamma(t))\,.</math>

By the [[Picard–Lindelöf theorem]], if <math>V</math> is [[Lipschitz continuity|Lipschitz continuous]] there is a ''unique'' <math>C^1</math>-curve <math>\gamma_x</math> for each point <math>x</math> in <math>S</math> so that, for some <math>\varepsilon > 0</math>,
<math display="block">\begin{align}
\gamma_x(0) &= x\\
\gamma'_x(t) &= V(\gamma_x(t)) \qquad \forall t \in (-\varepsilon, +\varepsilon) \subset \R.
\end{align}</math>

The curves <math>\gamma_x</math> are called '''integral curves''' or '''trajectories''' (or less commonly, flow lines) of the vector field <math>V</math> and partition <math>S</math> into [[equivalence class]]es. It is not always possible to extend the interval <math>(-\varepsilon,+\varepsilon)</math> to the whole [[real number line]]. The flow may for example reach the edge of <math>S</math> in a finite time.
<!--Integrating the vector field along any flow curve γ yields
<math display="block">\int_\gamma \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle = \int_a^b \langle \mathbf{F}( \boldsymbol{\gamma}(t) ), \boldsymbol{\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 <math>S</math>. If we drop a particle into this flow at a point <math>p</math> it will move along the curve <math>\gamma_p</math> in the flow depending on the initial point <math>p</math>. If <math>p</math> is a stationary point of <math>V</math> (i.e., the vector field is equal to the zero vector at the point <math>p</math>), then the particle will remain at <math>p</math>.

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 on <math>M</math> is called '''complete''' if each of its flow curves exists for all time.<ref>{{cite book |last=Sharpe | first= 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; it is described by a smooth mapping
:<math>\mathbf{R}\times M\to M.</math>
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 display="inline">x'(t) = x^2</math>, with initial condition <math>x(0) = x_0 </math>, has as its unique solution <math display="inline">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 \R</math> if  <math>x_0 = 0</math>).  Hence for <math>x_0 \neq 0</math>, <math>x(t)</math> is undefined at <math display="inline">t = \frac{1}{x_0}</math> so cannot be defined for all values of <math>t</math>.

===The Lie bracket===
The flows associated to two vector fields need not [[commutative property|commute]] with each other. Their failure to commute is described by the [[Lie bracket of vector fields|Lie bracket]] of two vector fields, which is again a vector field. The Lie bracket has a simple definition in terms of the action of vector fields on smooth functions <math>f</math>:
:<math>[X,Y](f):=X(Y(f))-Y(X(f)).</math>