TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Cartan's magic formula

Statement

For every vector field $X$ and differential form $ω$ , $L_{X} ω = d (ι_{X} ω) + ι_{X} (d ω) .$

Facts & Assumptions

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

[F1]

The preceding result states that For $f \in C^{\infty} (M) = Ω^{0} (M)$ , $d f (X) = X f$ for every smooth vector field $X$ . (The exterior derivative of a function is its differential).

[F2]

For a one-form $α$ , the invariant formula gives $d α (X, Y) = X α (Y) - Y α (X) - α ([X, Y])$ (The exterior derivative by the invariant vector-field formula).

Proof

technique · direct

1.1

For a function $f$ , the right side is $ι_{X} d f = X f = L_{X} f$ ; for a one-form $α$ , evaluate both sides on $Y$ and use $d α (X, Y) = X α (Y) - Y α (X) - α ([X, Y])$ .

F1F2given

2.1

Both $L_{X}$ and $d ι_{X} + ι_{X} d$ are degree-zero derivations, so equality on functions and exact coordinate one-forms extends to every local coordinate expression and hence globally.

step 1.1∎

Depends on

Used by

A closed form is flow-invariant when its contraction is zero Corollary
Lie derivative commutes with the exterior derivative Corollary
Cartan's formula for a coordinate vector field Example
Lie derivative of an area form and planar divergence Example
The Lie derivative is C^∞-linear in the vector field False statement
Cartan commutator identities Proposition
Lie derivatives are natural for related vector fields Proposition

Dependency tree · two levels

19 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)