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The divergence theorem for finite gluings of elementary solid regions

Statement

Let a finite gluing of elementary solid regions be given, with pieces $E_{1}, \dots, E_{N}$ , union $E$ and outer boundary presentation $Σ^{o u t}$ (Finite gluings of elementary solid regions and their outward boundary presentation), and let $F$ be a $C^{1}$ vector field on an open set containing $E$ . Then

$∭_{E} div F = \iint_{\partial E} ⟨ F, n ⟩,$

the right-hand side being the flux of $F$ over $Σ^{o u t}$ .

The decomposition into pieces, the internal-or-outer designation of the patches and the pairing of the internal patches are hypotheses supplied with the gluing; nothing is asserted about a solid presented without them.

Facts & Assumptions

Given: The finite gluing with its pieces $E_{1}, \dots, E_{N}$ , their presentations $Σ_{i}$ , the internal-or-outer designations, the pairing involution, the union $E$ , the outer presentation $Σ^{o u t}$ , and the $C^{1}$ field $F$ on an open $O \supseteq E$ .

[F1]

In a finite gluing the pieces are elementary solid regions with pairwise disjoint interiors whose union is $E$ , and the outer patches together form a compatible finite patch presentation of $\partial E$ (Finite gluings of elementary solid regions and their outward boundary presentation).

[F2]

The divergence of a $C^{1}$ field $F$ on an open subset of $R^{n}$ is $div F = \sum_{i < n} \partial_{i} F_{i}$ (Divergence and curl of a $C^{1}$ vector field).

[F3]

For a compatible finite patch presentation the oriented flux is the sum of the flux over its patches (Finitely patched regular surfaces, their area, scalar integrals, and flux), and $⟨ x, y ⟩ = \sum_{i < m} x_{i} y_{i}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L1]

For an elementary solid region $E^{'}$ with presentation $Σ^{'}$ and a $C^{1}$ field $F$ on an open set containing $E^{'}$ , $∭_{E^{'}} div F = \iint_{Σ^{'}} ⟨ F, n ⟩$ (The divergence theorem on an elementary solid region).

[L2]

For a finite gluing, $E$ is compact and Jordan measurable; for a continuous vector field on the union of the piece boundaries, the sum of the piece fluxes is the flux over the outer presentation; and for a continuous scalar function on $E$ , the sum of the piece volume integrals is the integral over the union (Internal faces cancel and volume integrals add when elementary solid regions are glued).

Proof

technique · direct

1.1

By [F1] each $E_{i}$ is an elementary solid region contained in $E$ , so $O$ is an open set containing $E_{i}$ and $F$ is $C^{1}$ on it; hence [L1] applies to each piece and gives $∭_{E_{i}} div F = \iint_{Σ_{i}} ⟨ F, n ⟩$ for $i = 1, \dots, N$ .

givenF1L1

1.2

The function $div F$ is continuous on $O$ by [F2], since a $C^{1}$ field has continuous first partial derivatives, and in particular continuous on $E$ .

givenF2

2.1

Summing the $N$ identities of step 1.1 over $i$ and applying [L2] to each side — the volume clause with $H = div F$ , continuous on $E$ by step 1.2, and the flux clause with $G = F$ , continuous on $\partial E$ and on every $\partial E_{i}$ — turns the left sum into $∭_{E} div F$ and the right sum into $\iint_{Σ^{o u t}} ⟨ F, n ⟩$ , which by [F1] and [F3] is the flux over the outer boundary presentation of $E$ .

step 1.1step 1.2F1F3L2

3.1

Step 2.1 is the asserted identity. The field is required to be $C^{1}$ on an open set containing the whole union, because step 1.1 applies the piecewise identity with that same field on each piece and step 2.1 integrates $div F$ over $E$ .

step 2.1∎

Remarks

What the gluing clause buys. A solid need not be simple in every coordinate direction: a U-shaped prism has sections in one direction that are unions of two disjoint intervals, so it admits no simple description there, and yet it is a gluing of three boxes. The companion examples page carries that computation.
No connectedness is used. The pieces need not touch and the boundary need not be connected: step 2.1 rearranges finitely many real numbers and integrates over a finite union.

Depends on

Used by

Dependency tree · two levels

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Sources

J. Feldman, A. Rechnitzer and E. Yeager, CLP-4 Vector Calculus (University of British Columbia), Theorem 4.2.2 (standard reference, not scraped)
G. Strang and E. Herman, Calculus Volume 3 (OpenStax), Theorem 6.20 (standard reference, not scraped)