False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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.
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Curve velocity coordinates depend on the chart

Statement

False claim: the coordinate tuple of a curve velocity is the same in every chart.

Facts & Assumptions

Given: On the first quadrant $Q : = {(x, y) : x > 0, y > 0}$ , the curve $γ (t) = (1 + t, 1)$ through $(1, 1)$ for $t$ near $0$ , together with the Cartesian chart and the polar chart $ψ (x, y) = (\sqrt{x^{2} + y^{2}}, \arctan (y / x)) .$

[L1]

Tangent coordinates change by the Jacobian of the coordinate transition (Change-of-coordinate formula for tangent bases).

Refutation

technique · direct

1.1

At $t = 0$ , the Cartesian velocity of $γ$ is $γ^{'} (0) = (1, 0) .$ In polar coordinates, $r (t) = \sqrt{(1 + t)^{2} + 1}, θ (t) = \arctan ⁣ (\frac{1}{1 + t}),$ so $r^{'} (0) = \frac{1}{\sqrt{2}}, θ^{'} (0) = - \frac{1}{2} .$ Thus the polar coordinate tuple of the same velocity is $(\frac{1}{\sqrt{2}}, - \frac{1}{2}) .$

givenalgebra

2.1

The two coordinate tuples from step 1.1 are not equal. This is consistent with [L1], which says they are related by the Jacobian of the coordinate change, not by identity.

L1step 1.1

3.1

Hence the coordinate tuple of a curve velocity need not be the same in two different charts. The statement is false.

step 1.1step 2.1∎

Depends on

Change-of-coordinate formula for tangent bases

Used by

Nothing in the library uses this result yet.

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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)