PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.

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Coordinate vector fields commute

Statement

In any smooth chart $(U, x^{1}, \dots, x^{n})$ , the coordinate vector fields $\partial / \partial x^{i}$ and $\partial / \partial x^{j}$ satisfy

$[\frac{\partial}{\partial x^{i}}, \frac{\partial}{\partial x^{j}}] = 0$

on $U$ .

Facts & Assumptions

Given: A smooth chart $(U, x^{1}, \dots, x^{n})$ and indices $i, j$ .

[L1]

The Lie bracket has the coordinate formula from the previous proposition (Coordinate formula for the Lie bracket).

Proof

technique · direct

1.1

In the chosen chart, the coefficient functions of $\partial / \partial x^{i}$ and $\partial / \partial x^{j}$ are constants: each is either $0$ or $1$ .

given

2.1

Substituting those constant coefficients into the formula of [L1] makes every derivative term vanish, so each coefficient of the bracket is zero.

L1step 1.1

3.1

Therefore the coordinate vector fields commute.

step 2.1∎

Depends on

Coordinate formula for the Lie bracket

Used by

Commuting coordinate fields have commuting flows Example
FALSE: a vanishing Lie bracket forces the vector fields to be pointwise linearly dependent False statement

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

Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)