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

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The exterior derivative is $C^{\infty}$ -linear

Statement

The assertion that $d (f ω) = f d ω$ for all smooth $f$ and forms $ω$ 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 homogeneous forms $α$ and $β$ , $d (α \land β) = d α \land β + (- 1)^{\deg α} α \land d β .$ (The exterior derivative is a graded derivation).

Refutation

technique · direct

1.1F1given

The graded Leibniz rule gives $d (f ω) = d f \land ω + f d ω$ .

2.1step 1.1∎

Take $M = R$ , $f = x$ , and $ω = 1$ ; then $d (f ω) = d x$ whereas $f d ω = 0$ .

Depends on

The exterior derivative is a graded derivation

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 exterior derivative is C∞C∞-linear