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# 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.

Related:

LeibnizNotationForVectorFields, CartanCalculus

Type of Math Object:

Definition

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Reference

## Mathematics Subject Classification

53-00*no label found*

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## Comments

## Missing notation

"T is a tensor" is too much vague, please specify the correct notation!