False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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The Lie derivative is $C^{\infty}$ -linear in the vector field

Statement

The assertion that $L_{f X} ω = f L_{X} ω$ for all smooth $f$ , $X$ , and $ω$ is false.

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that For every vector field $X$ and differential form $ω$ , $L_{X} ω = d (ι_{X} ω) + ι_{X} (d ω) .$ (Cartan's magic formula).

Refutation

technique · direct

1.1

Cartan's formula and $ι_{f X} = f ι_{X}$ give $L_{f X} ω = f L_{X} ω + d f \land ι_{X} ω$ .

F1given

2.1

On $R^{2}$ , take $f = x$ , $X = \partial_{y}$ , and $ω = d y$ ; the correction is $d x \neq 0$ .

step 1.1∎

Depends on

Cartan's magic formula

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)

The Lie derivative is C∞C∞-linear in the vector field