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A regular reparametrization of a connected parameter region has a constant orientation sign

Statement

Every regular reparametrization of a connected parameter region is either orientation-preserving everywhere or orientation-reversing everywhere.

Facts & Assumptions

Given: A regular reparametrization induced by a C1 diffeomorphism h between neighbourhoods of compact Jordan parameter regions whose interiors are nonempty and connected.

[L1]

If a C1 map has invertible derivative throughout a connected open set, then its Jacobian determinant is everywhere positive or everywhere negative (The Jacobian sign of a regular C1 map is constant on a connected domain).

[F1]

A reparametrization is orientation-preserving where det⁡Dh>0 and orientation-reversing where det⁡Dh<0 (Surface reparametrizations and their orientation sign).

Proof

technique · cases
1.1

Restrict h to the nonempty connected interior of the source region. Its derivative is invertible there, so [L1] gives a constant positive or negative determinant sign.

givenL1
2.1

Because the compact region is the closure of its interior and det⁡Dh is continuous and nonzero on a neighbourhood of it, the same strict sign holds on its boundary.

step 1.1given
3.1

In the positive case [assume-case pos], [F1] makes the reparametrization orientation-preserving everywhere; in the negative case [assume-case neg], [F1] makes it orientation-reversing everywhere. These cases exhaust [L1].

step 2.1F1cases-exhaustive∎

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