DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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.

Derivations at a point and the tangent space

Definition

Let $M$ be a smooth manifold and $p \in M$ . A derivation at $p$ is an $R$ -linear map $v : C_{p}^{\infty} (M) \to R$ such that $v ([f] [g]) = f (p) v ([g]) + g (p) v ([f])$ for all smooth germs $[f], [g] \in C_{p}^{\infty} (M)$ . The set of all derivations at $p$ is written $T_{p} M$ and called the tangent space of $M$ at $p$ .

Derivations at a point and the tangent space

Definition

Depends on

Dependency tree · 2 levels

Sources