DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

How 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 $C^{1}$ Euclidean map

Definition

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, and let $f : U \to R^{n}$ be $C^{1}$ . At a regular point, the derivative matrix is invertible (The regular locus of a square-dimensional $C^{1}$ 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 $C^{1}$ map is locally orientation-preserving where $\det D f > 0$ and locally orientation-reversing where $\det D f < 0$ . These are the two possible signs of the Jacobian determinant The Jacobian determinant of a square-dimensional $C^{1}$ map is the determinant of its Jacobian matrix at a regular point.

Depends on

Used by

Dependency tree · two levels

23 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. Lebl, Basic Analysis II, §8.5 (standard reference, not scraped)

Local orientation of a regular C1C1 Euclidean map