Let $M$ be a smooth manifold, $X$ a vector field on $M$ , and $T$ a tensor on $M$ . Then the Lie derivative $\mc{L}_XT$ of $T$ along $X$ is a tensor of the same rank as $T$ defined as $$\mc{L}_XT=\frac{d}{dt}\left(\rho^*_t(T)\right)|_{t=0}$$ 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 $\mc{L}_XY=[X,Y]$ , the standard Lie bracket of vector fields.