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.

Complete vector fields

Definition

A smooth vector field $X$ on $M$ is complete if, for every $p \in M$ , the maximal integral curve $γ_{p}$ of $X$ is defined on all of $R$ .

Depends on

Through each point there is a unique maximal integral curve

Used by

Local and global flows generated by a vector field Definition
The vector field x² d/dx has finite-time escape Example
FALSE: every smooth vector field is complete False statement
A vector field is complete if and only if its flow is global Proposition
Compactly supported smooth vector fields are complete Theorem

Dependency tree · two levels

3 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)
Will J. Merry, Differential Geometry (standard reference, not scraped)