TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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Surface area and scalar surface integrals are invariant under regular reparametrization

Statement

Surface area and scalar surface integrals are unchanged by every regular reparametrization, regardless of orientation sign.

Facts & Assumptions

Given: A regular reparametrization $ψ = φ \circ h$ from parameter region $E$ onto parameter region $D$ , and a continuous scalar field $q$ on the common patch image.

[L1]

Scalar surface integrals are parameter integrals of $(q \circ φ) J_{φ}$ , and $J_{ψ} = (J_{φ} \circ h) ∣ \det D h ∣$ (Surface area and scalar surface integrals on a regular patch, The oriented area vector transforms by the parameter Jacobian determinant).

[L2]

Compact-Jordan change of variables gives $\int_{E} (k \circ h) ∣ \det D h ∣ = \int_{D} k$ for a neighbourhood diffeomorphism carrying $E$ onto $D$ (Change of variables for an injective $C^{1}$ map on a compact Jordan set).

Proof

technique · direct

1.1

By [L1], the integral computed with $ψ$ is $\int_{E} (q \circ φ \circ h) (J_{φ} \circ h) ∣ \det D h ∣$ .

givenL1

2.1

Apply [L2] to $k = (q \circ φ) J_{φ}$ . The result is $\int_{D} (q \circ φ) J_{φ}$ , the integral computed with $φ$ .

step 1.1L2

3.1

The absolute determinant makes the calculation independent of orientation sign. Setting $q = 1$ gives invariance of area.

step 2.1L1∎

Depends on

Used by

Opposite parametrizations preserve area and negate flux Example

Dependency tree · two levels

23 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)