DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A smooth vector field is a smooth section of the tangent bundle

Definition

Assume ${A C}_{ω}$ , so that $T M$ carries its canonical smooth structure. Let $M$ be a smooth manifold. A smooth vector field on $M$ is a smooth section

$X : M \to T M$

of the tangent-bundle projection $π : T M \to M$ . Thus $π \circ X = {id}_{M}$ .

Equivalently, for each $p \in M$ the value $X (p)$ is a tangent vector in $T_{p} M$ , and the dependence on $p$ is smooth with respect to the canonical smooth structure on $T M$ .

Depends on

Used by

F-related vector fields Definition
Integral curves of a vector field Definition
The action of a vector field on smooth functions Definition
Time-dependent vector fields and their evolution operators Definition
FALSE: every pointwise assignment of a tangent vector is a smooth vector field False statement
A vector field along an embedded submanifold extends to a neighbourhood and globally when the submanifold is closed Lemma
A vector field tangent to an embedded submanifold restricts to a vector field on it Proposition
Smoothness of a vector field is equivalent to smooth coordinate components Proposition
Compactly supported smooth vector fields are complete Theorem
Derivations of smooth functions are exactly smooth vector fields Theorem

Dependency tree · two levels

11 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, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)