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.

The differential of a smooth map

Definition

Let $F : M \to N$ be smooth and let $p \in M$ . The differential of $F$ at $p$ is the map $d F_{p} : T_{p} M \to T_{F (p)} N$ defined by $d F_{p} (v) ([g]) : = v ([g \circ F])$ for every derivation $v \in T_{p} M$ and every germ $[g] \in C_{F (p)}^{\infty} (N)$ .

Depends on

Derivations at a point and the tangent space

Used by

Pullback of a cotangent vector Definition
The differential of a smooth real-valued function Definition
The global differential or tangent map Definition
The differential of a constant map is zero Example
The differential is intrinsic, not merely a Jacobian matrix False statement
Pullback of a target germ by a smooth map is a well-defined source germ Lemma
The differential sends derivations to derivations and is linear Lemma
Canonical tangent and cotangent splittings for products Theorem
Coordinate formula for the differential Theorem
The chain rule for differentials of smooth maps Theorem
The differential sends curve velocities to composite curve velocities Theorem

Dependency tree · two levels

3 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)