False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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A smooth Euclidean vector field need not be complete

Statement

False claim: every smooth vector field on $R^{n}$ is complete.

Facts & Assumptions

Given: The scalar autonomous ODE $x^{'} = x^{2}$ on $R$ with initial value $x (0) = 1$ .

[F1]

This is an autonomous ODE in the sense of Autonomous ordinary differential equations.

[L1]

Smooth autonomous ODEs have unique local smooth solutions (The fundamental theorem for autonomous smooth ODEs).

Refutation

technique · direct

1.1

The vector field $V (x) = x^{2}$ is smooth on $R$ . The explicit curve [F1, L1] $x (t) = 1 / (1 - t)$ satisfies $x (0) = 1$ and $x^{'} (t) = 1 / (1 - t)^{2} = x (t)^{2}$ for $t < 1$ , so [L1] identifies it with the unique local solution through $1$ .

F1L1

2.1

This solution cannot be extended past $t = 1$ as a real-valued solution, [step 1.1] because $x (t) \to + \infty$ as $t ↑ 1$ . Hence the maximal solution is not defined for all time.

step 1.1

3.1

Therefore a smooth Euclidean vector field need not be complete.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Nigel Hitchin, Differentiable Manifolds, Appendix §10.2 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.4 (standard reference, not scraped)