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

1.1

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 $d x^{ℓ} \land d x^{j} = - d x^{j} \land d x^{ℓ}$ .

F2givenalgebra

2.1

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

F1step 1.1∎

Depends on

Used by

Lie derivative commutes with the exterior derivative Corollary
Existence and uniqueness of the exterior derivative 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)