Hello all, I was hoping someone would be able to clarify this issue I am having with the Lie Derivative of a vector field.(adsbygoogle = window.adsbygoogle || []).push({});

We define the lie derivative of a vector field [itex]Y[/itex] with respect to a vector field [itex]X[/itex] to be

[itex]L_X Y :=\operatorname{\frac{d}{dt}} |_{t=0} (\phi_t^*Y)[/itex], where [itex]\phi_t[/itex] is the flow of [itex]X[/itex]. Now how do I know this thing is differentiable? And also, I'm not sure why this thing is a vector field on [itex]M[/itex] because what if our flow is only defined on a smaller open set [itex] V \subseteq M[/itex], and then for [itex] p \in M\setminus V[/itex] surely [itex](L_X Y)_p[/itex] doesn't make sense as a vector in [itex]T_p M[/itex] ?

Anyway thank you, any help would be appreciated.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Confusion over the definition of Lie Derivative of a Vector Field

Loading...

Similar Threads for Confusion definition Derivative |
---|

I Diffeomorphism invariance and contracted Bianchi identity |

I Basic Q about Vector/tensor definition and velocity |

I What is the covariant derivative of the position vector? |

I Definition of tangent space: why germs? |

A Intution behind the definition of extrinsic curvature |

**Physics Forums | Science Articles, Homework Help, Discussion**