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 of a form by a vector field

Definition

Let $X : M \to T M$ be a smooth section of the tangent bundle, that is, a smooth vector field on $M$ , and let $ω \in Ω^{k} (M)$ . The interior product $ι_{X} ω \in Ω^{k - 1} (M)$ is defined pointwise by

$(ι_{X} ω)_{p} : = ι_{X_{p}} ω_{p} .$

For $k = 0$ , adopt the formal convention $Ω^{- 1} (M) : = {0}$ and set $ι_{X} ω : = 0$ .

This pointwise field is smooth. Indeed, in local coordinates write $X = \sum_{j} X^{j} \partial_{j}$ and expand $ω$ in the smooth frame $d x^{I}$ as in Local coordinate expression for a differential form. The coefficients of $ι_{X} ω$ are signed finite sums of products $X^{j} ω_{I}$ , and are therefore smooth.

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed.