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Lie derivative (Definition)

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.




"Lie derivative" is owned by rspuzio. [ owner history (1) ]
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See Also: Leibniz notation for vector fields, Cartan calculus


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Lie derivative (for vector fields) (Definition) by matte
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Cross-references: Lie bracket, directional derivative, pullback, flow, rank, tensor, vector field, smooth manifold
There are 4 references to this entry.

This is version 3 of Lie derivative, born on 2002-12-10, modified 2006-10-15.
Object id is 3707, canonical name is LieDerivative.
Accessed 9956 times total.

Classification:
AMS MSC53-00 (Differential geometry :: General reference works )

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Missing notation by Mazzu on 2006-08-22 12:22:01
"T is a tensor" is too much vague, please specify the correct notation!
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