Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)
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FALSE: the point values X_p and Y_p determine the bracket value [X,Y]_p

Statement

False claim: if two pairs of vector fields agree pointwise at p, then they have the same Lie bracket value at p.

Facts & Assumptions

Given: On R, the pairs (X,Y)=(d/dx,x d/dx) and (X′,Y′)=(d/dx,0) at the point p=0.

[L1]

The Lie bracket acts on smooth functions by the commutator [X,Y]h=X(Yh)−Y(Xh) (The Lie bracket of smooth vector fields).

Refutation

technique · direct
1.1given

At p=0, both pairs have the same point values: X0=X0′=d/dx∣0 and Y0=Y0′=0.

1.2L1givenalgebra

For every smooth h:R→R, [L1] gives [X,Y]h=(xh′)′−xh′′=h′, while [X′,Y′]h=0. Thus [X,Y]0=d/dx∣0≠0=[X′,Y′]0.

2.1step 1.1step 1.2∎

Therefore the point values Xp and Yp do not determine the bracket value at p.

Depends on

Used by

Dependency tree · two levels

2 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