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Content-zero parameter-boundary exceptions do not affect surface integrals

Statement

Changing a bounded scalar or flux parameter integrand only on the content-zero boundary of a compact Jordan parameter region preserves integrability and its integral.

In both directions, one of the two bounded functions is integrable if and only if the other is, and then their integrals agree.

Facts & Assumptions

Given: A compact Jordan parameter region $D$ and bounded functions $p, q : D \to R$ that agree on $D^{\circ}$ .

[L1]

The boundary $\partial D$ has content zero (Regular parametrized surface patches on compact Jordan parameter regions, A bounded set in $R^{m}$ is Jordan measurable iff its boundary is null, equivalently of content zero).

[L2]

Bounded functions differing only on a content-zero set are integrable simultaneously and have equal integrals (Changing a bounded integrand on a content-zero set does not change its Riemann integral).

Proof

technique · direct

1.1

Since $p = q$ on $D^{\circ}$ , the set on which they differ is contained in $D ∖ D^{\circ} = \partial D$ , which has content zero by [L1].

givenL1

2.1

Apply [L2] to obtain both implications of the integrability equivalence and equality of the integrals.

step 1.1L2

3.1

A flux parameter integrand is scalar after taking the dot product, so the same argument applies to it. The conclusion licenses boundary seams, poles, and endpoint degeneracies only, not an interior rank failure.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources

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