DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

A smooth vector field is a smooth section of the tangent bundle

Definition

Assume ${A C}_{ω}$ , so that $T M$ carries its canonical smooth structure. Let $M$ be a smooth manifold. A smooth vector field on $M$ is a smooth section

$X : M \to T M$

of the tangent-bundle projection $π : T M \to M$ . Thus $π \circ X = {id}_{M}$ .

Equivalently, for each $p \in M$ the value $X (p)$ is a tangent vector in $T_{p} M$ , and the dependence on $p$ is smooth with respect to the canonical smooth structure on $T M$ .

A smooth vector field is a smooth section of the tangent bundle

Definition

Depends on

Dependency tree · 2 levels

Sources