TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Flux is invariant under orientation-preserving reparametrization and changes sign under reversal

Statement

An orientation-preserving reparametrization preserves flux and an orientation-reversing reparametrization negates it.

Facts & Assumptions

Given: A regular reparametrization $ψ = φ \circ h$ between connected parameter regions and a continuous vector field $F$ .

[L1]

Flux is the integral of $F \circ φ$ dotted with the oriented area vector, and that vector transforms by the signed factor $\det D h$ (Unit normal fields, orientations, and flux through a regular surface patch, The oriented area vector transforms by the parameter Jacobian determinant).

[L2]

The determinant has one constant sign on the parameter region (A regular reparametrization of a connected parameter region has a constant orientation sign), and compact-Jordan change of variables uses $∣ \det D h ∣$ (Change of variables for an injective $C^{1}$ map on a compact Jordan set).

Proof

technique · cases

1.1

By [L1], flux computed with $ψ$ is $\int_{E} ((F \circ φ) \cdot (φ_{u} \times φ_{v})) \circ h \det D h$ .

givenL1

2.1

In the preserving case [assume-case pos], [L2] gives $\det D h = ∣ \det D h ∣$ , so change of variables makes step 1.1 equal to the flux computed with $φ$ .

step 1.1L2

2.2

In the reversing case [assume-case neg], [L2] gives $\det D h = - ∣ \det D h ∣$ , so change of variables makes step 1.1 the negative of the flux computed with $φ$ .

step 1.1L2

3.1

The two constant-sign cases are exhaustive, proving both assertions.

step 2.1step 2.2cases-exhaustive∎

Depends on

Used by

Opposite parametrizations preserve area and negate flux Example
FALSE: flux is independent of the parametrization without an orientation condition False statement

Dependency tree · two levels

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Sources

University of Toronto MAT237 notes, Section 5.3, An Invariance Property (standard reference, not scraped)