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 rank of a derivative and constant-rank Euclidean maps
Definition
Let , let be open, and let be ( Euclidean maps and diffeomorphisms). The rank of at is where rank is the dimension of the image (Rank and nullity of a linear map with finite-dimensional domain). In the standard bases, this is also the rank of the Jacobian matrix (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).
For and , the map has constant rank on when for every . On the empty set this condition is vacuous, so it may hold for more than one ; every assertion that needs a determined rank will assume is nonempty or specify .
Depends on
Used by
- Submersions and immersions between Euclidean open sets Definition
- The map (x,y)↦(x,xy) has nonconstant rank on every neighbourhood of the origin Example
- A nonzero rank minor supplies the source coordinates for the constant-rank theorem Lemma
- Differential rank is lower semicontinuous Theorem
- The Euclidean constant-rank normal form Theorem
Dependency tree · two levels
15 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
- J. M. Lee, Introduction to Smooth Manifolds, Theorems 7.13 and 8.8-8.12 (standard reference, not scraped)
- L. W. Tu, An Introduction to Manifolds, Sections 11.1-11.2 (standard reference, not scraped)