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The volume of a glued elementary solid is a third of the outward flux of the position field

Statement

Let a finite gluing of elementary solid regions be given, with union $E$ and outer boundary presentation $Σ^{o u t}$ , and let $P$ be the position field $P (p) = p$ on $R^{3}$ . Then the content of the solid is a third of the outward flux of the position field through its boundary:

$cont (E) = \frac{1}{3} \iint_{\partial E} ⟨ P, n ⟩ .$

Moreover each of the three single-coordinate fields $p \mapsto p_{x} e_{x}$ , $p \mapsto p_{y} e_{y}$ and $p \mapsto p_{z} e_{z}$ satisfies

$\iint_{\partial E} ⟨ p_{k} e_{k}, n ⟩ = cont (E) (k \in {x, y, z}) .$

Facts & Assumptions

Given: The finite gluing with union $E$ and outer presentation $Σ^{o u t}$ , and the position field $P$ .

[F1]

The divergence of a $C^{1}$ field is $div G = \sum_{i < n} \partial_{i} G_{i}$ (Divergence and curl of a $C^{1}$ vector field).

[F2]

For $f = 1$ the zero extension is $1_{E}$ , so $\int_{E} 1 = cont (E)$ (The Riemann integral of a bounded function over a bounded Jordan measurable set, A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content).

[F3]

For $x, y \in R^{m}$ , $⟨ x, y ⟩ = \sum_{i < m} x_{i} y_{i}$ , and $e_{k}$ has $k$ th coordinate $1$ and the others $0$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ ).

[L1]

For a finite gluing with union $E$ and outer presentation $Σ^{o u t}$ and a $C^{1}$ field $G$ on an open set containing $E$ , $∭_{E} div G = \iint_{\partial E} ⟨ G, n ⟩$ (The divergence theorem for finite gluings of elementary solid regions).

[L2]

For a finite gluing, $E$ is compact and Jordan measurable (Internal faces cancel and volume integrals add when elementary solid regions are glued, Finite gluings of elementary solid regions and their outward boundary presentation).

[L3]

For integrable $f, g$ on a nondegenerate rectangle and scalars $α, β$ , the function $α f + β g$ is integrable with integral $α \int f + β \int g$ (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ).

Proof

technique · direct

1.1

The position field has $P_{k} (p) = p_{k}$ , so $\partial_{i} P_{k}$ is $1$ when $i = k$ and $0$ otherwise; these are continuous on $R^{3}$ , so $P$ is $C^{1}$ there and [F1] gives $div P = 1 + 1 + 1 = 3$ at every point. By [L2] the set $E$ is compact and Jordan measurable, so $\int_{E} 1 = cont (E)$ by [F2].

givenF1F2L2

1.2

For each $k$ the field $p \mapsto p_{k} e_{k}$ has $k$ th coordinate $p_{k}$ and the other two coordinates $0$ by [F3], so its only nonvanishing first partial derivative is $\partial_{k} p_{k} = 1$ ; it is therefore $C^{1}$ on $R^{3}$ with divergence $1$ by [F1].

givenF1F3

2.1

Applying [L1] with $G = P$ , which is $C^{1}$ on the open set $R^{3} \supseteq E$ , gives $\iint_{\partial E} ⟨ P, n ⟩ = \int_{E} div P = \int_{E} 3$ , and by [L3] with $α = 3$ , $β = 0$ and step 1.1 this is $3 \int_{E} 1 = 3 cont (E)$ . Dividing by $3$ gives the first identity.

step 1.1L1L3

2.2

Applying [L1] with $G$ the field $p \mapsto p_{k} e_{k}$ of step 1.2 gives $\iint_{\partial E} ⟨ p_{k} e_{k}, n ⟩ = \int_{E} 1 = cont (E)$ by step 1.1, for each of the three directions $k$ .

step 1.1step 1.2L1

3.1

Steps 2.1 and 2.2 are the asserted identities.

step 2.1step 2.2∎

Remarks

The solid is Jordan measurable because its pieces are, not by assumption. Step 1.1 takes that from [L2]; without it the symbol $cont (E)$ would not denote anything and $\int_{E} 1$ would not exist.
The orientation is what fixes the sign. Reversing every patch of the presentation negates the right-hand sides and would give a negative content. That the presentation of a glued elementary solid carries the outward normal is Every patch of an elementary solid region's presentation is a graph face in some direction, and at interior base points its normal is outward applied to each piece, and the companion examples page checks the sign against the known volume of a ball.

Depends on

Used by

The volume of a closed ball recovered from the outward flux of the position field Example

Dependency tree · two levels

68 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. Corral, Vector Calculus, chapter 4 (LibreTexts), Example 4.2 (standard reference, not scraped)
J. Feldman, A. Rechnitzer and E. Yeager, CLP-4 Vector Calculus (University of British Columbia), section 4.2 (standard reference, not scraped)