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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 $ \mathcal{L}_XT$ of $ T$ along $ X$ is a tensor of the same rank as $ T$ defined as

$\displaystyle \mathcal{L}_XT=\frac{d}{dt}\left(\rho^*_t(T)\right)\vert _{t=0}$
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}_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


Attachments:
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 5 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 8065 times total.

Classification:
AMS MSC: 53-00 (Differential geometry :: General reference works )
Pending Errata and Addenda
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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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