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Green's first identity on a glued elementary solid region

Statement

Let a finite gluing of elementary solid regions be given, with union $E$ and outer boundary presentation $Σ^{o u t}$ , let $O$ be an open set containing $E$ , let $u : O \to R$ be $C^{1}$ and let $v : O \to R$ be $C^{2}$ . Then

$∭_{E} (⟨ \nabla u, \nabla v ⟩ + u Δ v) = \iint_{\partial E} u ⟨ \nabla v, n ⟩,$

the right-hand side being the flux of the field $u \nabla v$ over $Σ^{o u t}$ .

No symmetry between $u$ and $v$ is claimed: the hypotheses on them differ.

Facts & Assumptions

Given: The finite gluing with union $E$ and outer presentation $Σ^{o u t}$ , the open $O \supseteq E$ , the $C^{1}$ function $u$ and the $C^{2}$ function $v$ on $O$ .

[F1]

For scalar-valued $f$ the gradient is $\nabla f = (\partial_{0} f, \dots, \partial_{m - 1} f)$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[F2]

For a $C^{2}$ function $f$ on an open subset of $R^{n}$ , $Δ f = div \nabla f = \sum_{i < n} \partial_{i} \partial_{i} f$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F3]

A scalar $f$ is of class $C^{k}$ on $U$ when every iterated derivative of length at most $k$ exists and is continuous on $U$ ( $C^{k}$ maps and multi-index derivative notation in Euclidean space), and a map is $C^{k}$ when each component is ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F4]

For $x, y \in R^{m}$ , $⟨ x, y ⟩ = \sum_{i < m} x_{i} y_{i}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and 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).

[L1]

Let $U \subseteq R^{n}$ be open, let $G : U \to R^{n}$ be $C^{1}$ and let $f : U \to R$ be $C^{1}$ . Then $f G$ is $C^{1}$ on $U$ and $div (f G) = ⟨ \nabla f, G ⟩ + f div G$ (Divergence and curl are linear and satisfy the scalar product rules).

[L2]

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).

Proof

technique · direct

1.1

Since $v$ is $C^{2}$ on $O$ , [F1] and [F3] make each component $\partial_{i} v$ of $\nabla v$ a function with continuous first partial derivatives, so $\nabla v$ is a $C^{1}$ field on $O$ .

givenF1F3

2.1

The function $u$ is $C^{1}$ on $O$ and $\nabla v$ is a $C^{1}$ field there by step 1.1, so [L1] with $f = u$ and $G = \nabla v$ makes $u \nabla v$ a $C^{1}$ field on $O$ with $div (u \nabla v) = ⟨ \nabla u, \nabla v ⟩ + u div \nabla v = ⟨ \nabla u, \nabla v ⟩ + u Δ v,$ the last equality by [F2] and [F4].

step 1.1F2F4L1

3.1

Applying [L2] to the $C^{1}$ field $u \nabla v$ on the open $O \supseteq E$ and substituting step 2.1 on the left gives $∭_{E} (⟨ \nabla u, \nabla v ⟩ + u Δ v) = \iint_{\partial E} ⟨ u \nabla v, n ⟩$ , and by [F4] the boundary integrand is $u ⟨ \nabla v, n ⟩$ . That is the asserted identity.

step 2.1F4L2∎

Remarks

The regularity is asymmetric because the identity is. The left-hand side applies $Δ$ to $v$ and only $\nabla$ to $u$ , so $v$ must be $C^{2}$ and $u$ need only be $C^{1}$ . Interchanging them is a different statement and needs $u$ to be $C^{2}$ as well; that is Green's second identity on a glued elementary solid region.
The boundary integrand is the normal derivative of $v$ , weighted by $u$ . The quantity $⟨ \nabla v, n ⟩$ is the derivative of $v$ in the direction of the boundary normal, and the identity says that its $u$ -weighted boundary integral is controlled by $Δ v$ and by the pairing of the two gradients inside the solid. Taking $u$ identically $1$ makes the first volume term vanish, which is the form used on the companion examples page.

Depends on

Used by

Dependency tree · two levels

33 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

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