CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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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.
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A compactly supported smooth Euclidean vector field is complete

Statement

Every compactly supported smooth vector field on $R^{n}$ is complete.

Facts & Assumptions

Given: A smooth vector field $V : R^{n} \to R^{n}$ whose support is contained in a compact set $K$ .

[L1]

A bounded locally Lipschitz vector field on all of Euclidean space is complete (A bounded vector field on all of Euclidean space is complete).

[L2]

A continuous real-valued function on a nonempty compact metric space attains its maximum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Proof

technique · direct

1.1

The norm function $x \mapsto ∥ V (x) ∥_{2}$ is continuous on the compact support [L2] $K$ , so [L2] gives a bound $M$ there. Outside $K$ the vector field vanishes by definition of support. Hence $∥ V (x) ∥_{2} \leq M$ for every $x \in R^{n}$ .

2.1

Smooth maps are locally Lipschitz on Euclidean open sets, so step 1.1 makes [L1, step 1.1] $V$ a bounded locally Lipschitz vector field. Therefore [L1] applies and yields completeness.

L1step 1.1∎

Depends on

Used by

A compactly supported vector field with global solutions Example

Dependency tree · two levels

26 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)