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

How statement and proof provenance work

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

1.1

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.

givenF1

2.1

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.

step 1.1F1algebra∎

Source notes

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

Depends on

First Green identity

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Hunter, Notes on Partial Differential Equations (standard reference, not scraped)