DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-11

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.

The Jacobian determinant of a square-dimensional $C^{1}$ map is the determinant of its Jacobian matrix

Definition

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, and let $g = (g_{1}, \dots, g_{n}) : U \to R^{n}$ be $C^{1}$ . Its Jacobian matrix at $x \in U$ is $D g (x) = {(\frac{\partial g_{i}}{\partial x_{j}} (x))}_{1 \leq i, j \leq n},$ the matrix of the total derivative in the standard bases. Its Jacobian determinant is $\det D g (x) .$ The change-of-variables scale factor is $∣ \det D g (x) ∣$ . Thus an orientation-reversing derivative and an orientation-preserving derivative with the same volume scale have the same change-of-variables factor.

Depends on

Dependency tree · 2 levels

Sources

A. Leibman, Multidimensional Real Analysis, §5.5

The Jacobian determinant of a square-dimensional C1C1 map is the determinant of its Jacobian matrix

Definition

Depends on

Dependency tree · 2 levels

Sources

The Jacobian determinant of a square-dimensional $C^{1}$ map is the determinant of its Jacobian matrix