CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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 nonvanishing vector field has locally parallel integral curves

Statement

Near any point where a smooth vector field does not vanish, its integral curves are the parallel coordinate lines of a flow-box chart.

Facts & Assumptions

Given: A smooth vector field $X$ and a point $p$ with $X_{p} \neq 0$ .

[L1]

Near $p$ there are coordinates in which $X = \partial / \partial u^{1}$ (The flow-box theorem).

Proof

technique · direct

1.1

In the coordinates given by [L1], the integral-curve equation for $X$ is ${\dot{u}}^{1} = 1, {\dot{u}}^{2} = \dots = {\dot{u}}^{n} = 0.$

L1given

2.1

Therefore the integral curves are exactly the lines $t \mapsto (t + c_{1}, c_{2}, \dots, c_{n}),$ which are parallel to the $u^{1}$ -axis.

step 1.1

3.1

So a nonvanishing vector field has locally parallel integral curves.

step 2.1∎

Depends on

The flow-box theorem

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)