DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-30

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 velocity derivation of a smooth curve

Definition

Let $M$ be a smooth manifold, let $p \in M$ , and let $γ$ be a smooth curve in $M$ with $γ (0) = p$ . Its velocity derivation at $0$ is the map $\dot{γ} (0) : C_{p}^{\infty} (M) \to R$ defined by $\dot{γ} (0) ([f]) : = (f \circ γ)^{'} (0) .$ This is representative independent because equal germ representatives agree on a neighbourhood of $p$ , so their composites with $γ$ agree near $0$ . Linearity and the Leibniz rule follow from the corresponding one-variable derivative rules, so $\dot{γ} (0)$ is a derivation at $p$ .

Depends on

The germ of a smooth function at a point

The velocity derivation of a smooth curve

Definition

Depends on

Dependency tree · 2 levels

Sources