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.

Time-dependent vector fields and their evolution operators

Definition

Assume ${A C}_{ω}$ , so that $T M$ carries its canonical smooth structure. Let $M$ be a smooth manifold, and let $I \subseteq R$ be an interval. A time-dependent vector field on $M$ over $I$ is a smooth map

$X : I \times M \to T M$

such that $X (t, p) \in T_{p} M$ for every $(t, p) \in I \times M$ . One often writes $X_{t} (p) : = X (t, p)$ .

An evolution operator for $X$ is a family of maps

$Ψ_{t, s} : U_{s} \to U_{t}$

defined for pairs $(t, s)$ in some time domain, such that for each $p \in U_{s}$ the curve $r \mapsto Ψ_{r, s} (p)$ solves

$\frac{d}{d r} Ψ_{r, s} (p) = X_{r} (Ψ_{r, s} (p)), Ψ_{s, s} (p) = p .$

Depends on

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

Time-dependent vector fields and their evolution operators

Definition

Depends on

Dependency tree · 2 levels

Sources