DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Interior product on alternating covectors

Definition

Let $V$ be a finite-dimensional real vector space, let $v \in V$ , and let $ω \in {Alt}^{k} (V)$ with $k \geq 1$ . The interior product $ι_{v} ω \in {Alt}^{k - 1} (V)$ is defined by

(ι_{v} ω) (v_{2}, \dots, v_{k}) : = ω (v, v_{2}, \dots, v_{k}) .

For $k = 0$ , adopt the formal convention ${Alt}^{- 1} (V) : = {0}$ and set $ι_{v} ω : = 0$ .

Depends on

Alternating $k$-covectors

Used by

Interior product of a form by a vector field Definition
Interior product is a graded antiderivation Proposition

Dependency tree · two levels

2 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)