DefinitionDefinition: AI-adaptedProof: 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.

Local and global flows generated by a vector field

Let $X$ be a smooth vector field on $M$ .

A local flow of $X$ consists of an open set $D \subseteq R \times M$ containing ${0} \times M$ and a smooth map $Φ : D \to M$ such that:

$Φ (0, p) = p$ for every $p \in M$ ;
for each $p$ , the fibre $D_{p} : = {t : (t, p) \in D}$ is an interval;
for each $p$ , the curve $t \mapsto Φ (t, p)$ is an integral curve of $X$ on $D_{p}$ ;
whenever both sides are defined, $Φ (t, Φ (s, p)) = Φ (t + s, p)$ .

If $D = R \times M$ , then $Φ$ is the global flow of $X$ .

Depends on