DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-06

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A graded derivation of the algebra of differential forms

Definition

Let $M$ be a smooth manifold and $r \in Z$ , and put $Ω^{j} (M) = {0}$ for $j < 0$ . A degree- $r$ graded derivation of $Ω^{*} (M)$ is an $R$ -linear map $D : Ω^{*} (M) \to Ω^{*} (M)$ such that $D Ω^{k} (M) \subseteq Ω^{k + r} (M)$ for every $k \geq 0$ and $D (α \land β) = D α \land β + (- 1)^{r \deg α} α \land D β$ for all homogeneous forms $α$ and $β$ .

Depends on

Used by

The exterior derivative is a graded derivation 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)