DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-04

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The action of a vector field on smooth functions

Definition

Let $X$ be a smooth vector field on a smooth manifold $M$ . Its action on smooth functions is the operator

$X : C^{\infty} (M) \to C^{\infty} (M)$

defined pointwise by

$(X f) (p) : = X_{p} (f)$

for every $f \in C^{\infty} (M)$ and $p \in M$ , where $X_{p} \in T_{p} M$ is viewed as a derivation at $p$ .

Depends on

Used by

The Lie bracket of smooth vector fields Definition
The Lie derivative of a function Definition
A vector field acts as a derivation of smooth functions Proposition
F-relatedness is equivalent to the derivation intertwining law Proposition
Derivations of smooth functions are exactly smooth vector fields Theorem

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

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