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.
Smooth functions on relatively open half-space sets
Definition
If is relatively open in , a map is smooth when every has an Euclidean-open neighbourhood of and a smooth with on . This is a local condition; maps into a relatively open half-space are smooth when their Euclidean coordinate functions are.
Depends on
Used by
- Smooth functions and tensor fields extend locally across the boundary Corollary
- Smooth charts, atlases, and structures with boundary Definition
- Smooth maps between manifolds with boundary Definition
- Half-space extensions agreeing on a relatively open set have the same derivatives there Lemma
- Chain rule for smooth half-space maps Proposition
- Boundary submanifolds of a boundaryless manifold have half-slice charts Theorem
Dependency tree · one level
1 result within one dependency step 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
- Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
- Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)