Editing Vector field (section)

=== 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>.