# Lie derivative

Let $M$ be a smooth manifold, $X$ a vector field^{} on $M$, and $T$ a tensor on $M$. Then the Lie derivative^{}
${\mathcal{L}}_{X}T$ of $T$ along $X$ is a tensor of the same rank as $T$ defined as

$${\mathcal{L}}_{X}T={\frac{d}{dt}\left({\rho}_{t}^{*}(T)\right)|}_{t=0}$$ |

where $\rho $ is the flow of $X$, and ${\rho}_{t}^{*}$ is pullback by ${\rho}_{t}$.

The Lie derivative is a notion of directional derivative^{} for tensors.
Intuitively, this is the change in $T$ in the direction of $X$.

If $X$ and $Y$ are vector fields, then ${\mathcal{L}}_{X}Y=[X,Y]$, the standard Lie bracket of vector fields.

Title | Lie derivative |
---|---|

Canonical name | LieDerivative |

Date of creation | 2013-03-22 13:14:10 |

Last modified on | 2013-03-22 13:14:10 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 53-00 |

Related topic | LeibnizNotationForVectorFields |

Related topic | CartanCalculus |