DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-24

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.

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Surface reparametrizations and their orientation sign

Definition

Let $(D, φ)$ and $(E, ψ)$ be regular surface patches. A regular reparametrization from $ψ$ to $φ$ is a $C^{1}$ diffeomorphism $h$ between neighbourhoods of their compact Jordan parameter regions with $ψ = φ \circ h$ . We additionally require $h [E] = D$ ( $C^{k}$ Euclidean maps and diffeomorphisms).

The derivative $D h$ is invertible, so its Jacobian determinant is nonzero (The Jacobian determinant of a square-dimensional $C^{1}$ map is the determinant of its Jacobian matrix). A regular surface reparametrization is orientation-preserving when its parameter Jacobian determinant is positive and orientation-reversing when it is negative. Constancy of the sign on a connected parameter region is proved separately; the terms here apply pointwise whenever needed.

Depends on

Used by

Dependency tree · two levels

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Sources

M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)
University of Toronto MAT237 notes, Section 5.3, An Invariance Property (standard reference, not scraped)