The Green identities used here are Euclidean

The integration pair is the Euclidean one. Every boundary flux computed on this page uses the Euclidean surface measure, the divergence identity Divergence on a bounded C1 Euclidean domain for a bounded $C^1$ domain with the outward normal, its piecewise-boundary relative Divergence for finite piecewise C1 presentations, and the two-function identity Second Green identity. No step of any item on this page integrates a differential form, and no step uses a Stokes theorem for oriented manifolds.

Where the pair is actually used. The distributional computation The negative Laplacian of the fundamental solution is the unit Dirac distribution and the Green representation formula Green representation for classical Poisson data excise the pole from the domain and then apply the second Green identity on the resulting bounded $C^1$ domain; the singular flux of the kernel across the excised sphere is computed directly from the radial profile and the chart surface integral of Surface integration on compact C1 hypersurfaces. The Neumann compatibility identity and the boundary terms of the corollaries use the divergence theorem in the same way.

Outward normals on excised balls are reversed. On the outer boundary the normal is the outward unit normal of $\mathrm{\Omega }$; on an excised sphere $S(x,\epsilon )$ the outward normal of the remaining domain $\mathrm{\Omega }\setminus \bar{B}(x,\epsilon )$ points into the hole, that is, $\sigma (y)=-(y-x)\mathrm{/}\mathrm{\mid }y-x\mathrm{\mid }$. Both cited items display this reversal, and the sign of every singular boundary term is read off from it. Nothing here depends on the orientation convention of a manifold boundary.

The later manifold Stokes theorem is context only. A general Stokes theorem for oriented manifolds is built later in this run, on a page that this pair does not require and that does not require this pair. It supplies no proof step, no hypothesis and no sign convention to any item here, and no statement proved here is claimed as a manifold statement. The orientation language of the Euclidean surface measure is self-contained at this point.

Choice. Every item that invokes the Euclidean integration pair states ${\mathrm{A}\mathrm{C}}_{\omega }$ and inherits it through the measure, polar-surface, divergence and Green-identity conventions listed above; The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is the only choice principle used on this page. No full Axiom of Choice and no incompatible-axiom branch occurs.

Hunter, Notes on Partial Differential Equations (2014), §§1.11–1.12, printed pp.16–18, proves the divergence theorem by graph integration, and §2.5, printed p.32, derives the Green identities from it; the surface measure used there is the Euclidean chart measure. Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript), §5.4 equations (5.31)–(5.36), printed pp.125–126, records the same Euclidean surface and Green identities in the sign convention $-\mathrm{\Delta }\mathrm{\Phi }={\delta }_0$ used here. Neither reference is used as a proof of the statements above: this remark only fixes which earlier local results carry every flux computation on the page, and records that the manifold comparison is not one of them.