# Lie derivative

Let $M$ be a smooth manifold, $X$ a vector field on $M$, and $T$ a tensor on $M$. Then the $\mathcal{L}_{X}T$ 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 LieDerivative 2013-03-22 13:14:10 2013-03-22 13:14:10 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Definition msc 53-00 LeibnizNotationForVectorFields CartanCalculus