 Euclidean scalar-gradient stress identity

ID: euclidean-scalar-gradient-stress-identity

Codex  0 Created 2026-10-18 Updated 2026-10-25

For a sufficiently smooth real scalar field on Euclidean space, set

Σ_{kj} = \partial_{k} ψ \partial_{j} ψ - δ_{kj} ∣\nabla ψ ∣^{2} /2

. The product rule and equality of mixed partial derivatives give

\partial_{k} Σ_{kj} = (Δ ψ) \partial_{j} ψ + \partial_{k} ψ \partial_{k} \partial_{j} ψ - \partial_{k} ψ \partial_{j} \partial_{k} ψ = (Δ ψ) \partial_{j} ψ .

The divergence theorem, applied twice, consequently gives the useful pure-boundary identity

\int_{V} r \nabla \cdot [(Δ ψ) \nabla ψ] d V = \int_{\partial V} [r (Δ ψ) \partial_{n} ψ - (\partial_{n} ψ) \nabla ψ + \frac{1}{2} ∣\nabla ψ ∣^{2} \hat{n}] d S .

Here

\partial_{n} ψ = \hat{n} \cdot \nabla ψ

uses the outward unit normal. This is the Euclidean zero-potential counterpart of the scalar stress-energy divergence identity.

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