DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Lie derivative of a vector field

Definition

Let $X$ be a smooth vector field with maximal flow $Φ$ , and let $Y$ be another smooth vector field on $M$ . The Lie derivative of $Y$ along $X$ is the vector field defined by

$(L_{X} Y)_{p} : = {\frac{d}{d t} ∣}_{t = 0} (Φ_{- t})_{*} Y_{Φ_{t} (p)} .$

The inverse-time pushforward is the sign convention that later yields $L_{X} Y = [X, Y]$ .

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)