Alphabeta MathHow 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 germ of a smooth function at a point
Definition
Let be a smooth manifold and . Two pairs and , where and are open neighbourhoods of and , are smooth, are equivalent at when there is an open neighbourhood of such that . An equivalence class is a germ of a smooth function at .
Depends on
Used by
- Coordinate derivations at a point Definition
- The local algebra of smooth function germs Definition
- The velocity derivation of a smooth curve Definition
- Algebra operations on smooth germs are representative independent Lemma
- Coordinate derivations are well-defined derivations Lemma
- Equality near a point is an equivalence relation on local smooth functions Lemma
- Pullback of a target germ by a smooth map is a well-defined source germ Lemma
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)