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.