DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Lie derivative of a tensor field

Definition

If $X$ has local flow $Φ_{t}$ and $T$ is a smooth tensor field of type $(r, s)$ , its Lie derivative is $L_{X} T = {\frac{d}{d t} ∣}_{t = 0} Φ_{t}^{*} T,$ on every local flow domain where this derivative is defined. Here the pullback by the local diffeomorphism $Φ_{t}$ acts by $d Φ_{- t}$ on each contravariant slot and by $d Φ_{t}$ on each covariant slot.

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed.