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.

Coordinate derivations at a point

Definition

Let $(U, x)$ be a smooth chart on an $n$ -manifold $M$ with $p \in U$ and $a : = x (p)$ . For $1 \leq i \leq n$ , the $i$ th coordinate derivation at $p$ is the map $\partial_{i} ∣_{p} : C_{p}^{\infty} (M) \to R$ defined by $\partial_{i} ∣_{p} ([f]) : = \frac{\partial (f \circ x^{- 1})}{\partial x^{i}} (a) .$ This is the intrinsic version of differentiating in the $i$ th chart direction.

Coordinate derivations at a point

Definition

Depends on

Dependency tree · 2 levels

Sources