CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Second Green identity

Statement

Assume ${A C}_{ω}$ . For a bounded $C^{1}$ domain $Ω \subset R^{n}$ , $n \geq 2$ , or the specified finite piecewise $C^{1}$ class, and real $u, v \in C^{2} (\overline{Ω})$ , $\int_{Ω} (v Δ u - u Δ v) d x = \int_{\partial Ω} (v \partial_{ν} u - u \partial_{ν} v) d S .$ The face convention and finiteness are those of the first Green identity; every normal is outward from Omega, including normals on holes.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , a bounded C1 domain or the specified finite piecewise C1 class in dimension $n \geq 2$ , and real $u, v \in C^{2} (\overline{Ω})$ .

[F1]

The first identity applies to each ordered pair of C2 functions. (First Green identity).

Proof

1.1givenF1

Both functions are C1 as well as C2, so apply F1 to (u,v) and (v,u). This gives $\int_{Ω} (v Δ u + D u \cdot D v) = \int_{\partial Ω} v \partial_{ν} u$ and $\int_{Ω} (u Δ v + D v \cdot D u) = \int_{\partial Ω} u \partial_{ν} v$ , using the same specified outward normal and faces in both applications.

2.1step 1.1F1algebra∎

All four integrals in step 1.1 are finite by F1, so subtraction is legitimate. Since $D u \cdot D v = \sum_{i} (\partial_{i} u) (\partial_{i} v) = D v \cdot D u$ pointwise, those terms cancel. Linearity leaves exactly the asserted volume and boundary differences, with the unchanged outward normals on every component.

Source notes

Hunter §2.5, Theorem 2.23, equation (2.12) and proof, printed p. 32 (PDF p. 38).

Second Green identity

Statement

Facts & Assumptions

Proof

Source notes

Depends on

Dependency tree · 2 levels

Sources