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

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The exterior derivative squares to zero

Statement

For every differential form $ω$ , $d (d ω) = 0$ .

Facts & Assumptions

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

[F1]

The preceding result states that If $ω = η$ on an open set $U$ , then $d ω = d η$ on $U$ . (The exterior derivative is local).

[F2]

Smooth coefficient functions have equal mixed second partial derivatives (Clairaut--Schwarz theorem for continuous second partial derivatives).

Proof

technique · direct

1.1F2givenalgebra

On a chart, write $ω = \sum_{I} ω_{I} d x^{I}$ . Applying the coordinate formula twice gives $d^{2} ω = \sum_{I, j, ℓ} \partial_{ℓ} \partial_{j} ω_{I} d x^{ℓ} \land d x^{j} \land d x^{I} .$ The terms with $j = ℓ$ vanish. For $j \neq ℓ$ , the terms indexed by $(j, ℓ)$ and $(ℓ, j)$ cancel because mixed partials of the smooth coefficient $ω_{I}$ agree while $d x^{ℓ} \land d x^{j} = - d x^{j} \land d x^{ℓ}$ .

2.1F1step 1.1∎

The coordinate identity holds on every chart and therefore globally by locality.

Depends on

Used by

Lie derivative commutes with the exterior derivative Corollary
Closed and exact differential forms Definition
De rham cochain complex Definition
Every smooth form is either closed or exact False statement
Curvature of an extending Bott connection lies in the transverse differential ideal Lemma
Independence of the auxiliary form eta up to exact forms Lemma
Rescaling the defining form changes the Godbillon-Vey form by an exact form Lemma
The de Rham complex and pullback extend to manifolds with boundary Lemma
The Godbillon-Vey form eta wedge d eta is closed Lemma
Existence and uniqueness of the exterior derivative Theorem
Harmonic representatives for compact de Rham and Dolbeault complexes Theorem
Second Bianchi identity for a bundle connection Theorem
The d, partial and dbar identities Theorem

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

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