Alphabeta MathHow statement and proof provenance work
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Surface reparametrizations and their orientation sign
Definition
Let and be regular surface patches. A regular reparametrization from to is a diffeomorphism between neighbourhoods of their compact Jordan parameter regions with . We additionally require ( Euclidean maps and diffeomorphisms).
The derivative is invertible, so its Jacobian determinant is nonzero (The Jacobian determinant of a square-dimensional 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
- A regular reparametrization of a connected parameter region has a constant orientation sign Corollary
- Finite gluings of elementary solid regions and their outward boundary presentation Definition
- A U-shaped prism is a finite gluing of three boxes and is not simple in every coordinate direction Example
- FALSE: the patches of a finite presentation can always be reoriented to make their normals agree on overlaps False statement
- Internal faces cancel and volume integrals add when elementary solid regions are glued Lemma
- The oriented area vector transforms by the parameter Jacobian determinant Lemma
- The tangent plane is invariant under regular reparametrization Proposition
Dependency tree · two levels
14 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
- 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)