Lie derivative

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

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

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 ${ℒ}_{X}Y=[X,Y]$, 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