PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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Lie derivative of forms is a degree-zero graded derivation

Statement

For forms $α, β$ , $L_{X} (α \land β) = (L_{X} α) \land β + α \land (L_{X} β)$ .

Facts & Assumptions

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

[F1]

The preceding result states that For $ω \in Ω^{k} (M)$ , $L_{X} ω$ is the Lie derivative of $ω$ regarded as an alternating covariant tensor. (The Lie derivative of a differential form).

Proof

technique · direct

1.1

The wedge of alternating forms is the alternating restriction of their tensor product.

F1given

2.1

Restricting the tensor product rule for $L_{X}$ to that alternating product yields the degree-zero graded rule.

step 1.1∎

Depends on

Used by

Cartan's magic formula Theorem

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)