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.
Local orientation of a regular Euclidean map
Definition
Let , let be open, and let be . At a regular point, the derivative matrix is invertible (The regular locus of a square-dimensional map, A square matrix is invertible exactly when its multiplication map is a linear isomorphism; matrices preserve inverses of linear isomorphisms), so its determinant is a nonzero real number (An invertible square matrix over a commutative ring has unit determinant, Ordered field).
A regular map is locally orientation-preserving where and locally orientation-reversing where . These are the two possible signs of the Jacobian determinant The Jacobian determinant of a square-dimensional map is the determinant of its Jacobian matrix at a regular point.
Depends on
The Jacobian determinant of a square-dimensional map is the determinant of its Jacobian matrix
The regular locus of a square-dimensional map
Ordered field
A square matrix is invertible exactly when its multiplication map is a linear isomorphism; matrices preserve inverses of linear isomorphisms
An invertible square matrix over a commutative ring has unit determinant