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 $p$ .

Depends on

Used by

Coordinate derivations at a point Definition
The differential of a smooth map Definition
The tangent bundle as a disjoint union Definition
A tangent vector is not an ambient arrow by definition False statement
Linearity alone does not make a tangent vector False statement
The tangent space is intrinsically defined False statement
A derivation annihilates constant germs Lemma
Coordinate derivations are well-defined derivations Lemma
The differential sends derivations to derivations and is linear Lemma

Dependency tree · two levels

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