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

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First Green identity

Statement

Assume ${A C}_{ω}$ . Let $Ω \subset R^{n}$ , $n \geq 2$ , be a bounded $C^{1}$ domain or have the specified finite piecewise $C^{1}$ presentation. For real $u \in C^{2} (\overline{Ω})$ and $v \in C^{1} (\overline{Ω})$ , $\int_{Ω} (v Δ u + D u \cdot D v) d x = \int_{\partial Ω} v \partial_{ν} u d S .$ In the piecewise case the right side is the sum over faces, counted once off E. All integrals are finite.

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 $u \in C^{2} (\overline{Ω})$ , $v \in C^{1} (\overline{Ω})$ .

[F1]

The divergence formula holds for a C1 field up to a C1 boundary. (Divergence on a bounded C1 Euclidean domain).

[F2]

The divergence formula holds for the specified finite faces. (Divergence for finite piecewise C1 presentations).

[F3]

The normal derivative is the gradient dotted with the outward normal. (Classical normal derivative).

[F4]

The product rule applies to scalar derivatives. (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Proof

1.1

Set $F_{i} = v \partial_{i} u$ . On each coordinate line inside Omega the product rule F4 gives $\partial_{i} F_{i} = (\partial_{i} v) (\partial_{i} u) + v \partial_{i}^{2} u$ ; the same rule for every $\partial_{j} F_{i}$ shows that F and all its first derivatives extend continuously to the closure. Thus $F \in C^{1} (\overline{Ω}; R^{n})$ and summing over i gives $div F = D u \cdot D v + v Δ u$ .

givenF4algebra

2.1

On every regular boundary point F3 gives $F \cdot ν = v (D u \cdot ν) = v \partial_{ν} u$ . Apply F1 in the C1 case and F2 in the specified piecewise case to the field from step 1.1. Substituting its divergence and flux gives exactly the displayed identity. Continuous derivatives and functions on the compact closure are bounded; finite volume and the finite surface measures in F1/F2 prove absolute integrability. The edge set contributes zero to each face integral.

step 1.1F1F2F3algebra∎

Source notes

Hunter §2.5, Theorem 2.23, equation (2.11) and its proof, printed p. 32 (PDF p. 38); the weaker C1 assumption on v follows from the displayed product computation.

Depends on

Used by

Second Green identity Corollary

Dependency tree · two levels

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Sources

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