PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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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 is the commutator of vector-field actions on smooth functions (The Lie bracket of smooth vector fields).

[L2]

Mixed coordinate partial derivatives of a smooth scalar function commute (Clairaut--Schwarz theorem for continuous second partial derivatives).

Proof

technique · direct

1.1given

In the chosen chart, the coordinate fields act on smooth functions by $\partial_{i}$ and $\partial_{j}$ .

2.1L1L2step 1.1

For every smooth function $f$ on the chart, [L1] and [L2] give $[\partial_{i}, \partial_{j}] f = \partial_{i} \partial_{j} f - \partial_{j} \partial_{i} f = 0$ . Thus the bracket is the zero derivation, hence the zero coordinate vector field.

3.1step 2.1∎

Therefore the coordinate vector fields commute.

Depends on

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
Coordinate formula for the curvature tensor Proposition
The coordinate formula for the Lie derivative of a covariant tensor Proposition
The local coordinate formula for the exterior derivative Theorem

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)