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.
Half-space extensions agreeing on a relatively open set have the same derivatives there
Statement
If two smooth Euclidean extensions agree on a relatively open subset of , then all of their derivatives agree at every point of that subset.
Facts & Assumptions
Given: A relatively open set and two smooth Euclidean maps and , defined on neighbourhoods of , whose restrictions to agree.
Proof
Their difference vanishes on the relative interior, which is dense in the relative set. Every Euclidean derivative of the difference vanishes there by ordinary differentiation.
Those derivatives are continuous, so they also vanish at each face point. Hence the derivative of a half-space-smooth map is independent of its chosen extension.
Depends on
Used by
- Boundary smoothness is independent of charts and extensions Lemma
- Form calculus extends locally across a manifold boundary Lemma
- Local side-preserving extensions of half-space transitions Lemma
- The de Rham complex and pullback extend to manifolds with boundary Lemma
- Chain rule for smooth half-space maps Proposition
- Tangent and cotangent bundles extend over a boundary Theorem
Dependency tree · two levels
2 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
- 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)