Divergence for finite piecewise C1 presentations

For epsilon decreasing to zero use F2 and set $F_{\epsilon }=(1-{\eta }_{\epsilon })F$. Its support in the compact closure avoids a neighborhood of E. The remaining boundary support has a finite cover by the single-graph neighborhoods of F1; add an interior open set. F3 partitions this support and F4 applies to each localized term. Summing the product rules gives ${\int }_{\mathrm{\Omega }}\mathrm{div}{F}_{\epsilon }={\sum }_j{\int }_{S_j}(1-{\eta }_{\epsilon })F\cdot {\nu }_{j}dS$. Null overlaps in F1 prevent double counting.

The product rule gives $\mathrm{div}{F}_{\epsilon }=(1-{\eta }_{\epsilon })\mathrm{div}F-D{\eta }_{\epsilon }\cdot F$. The omitted first term has integral bounded by $\mathrm{\parallel }\mathrm{div}F{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }\mathrm{supp}{\eta }_{\epsilon }\mathrm{\mid }\to 0$ and the second by $\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty }}\int \mathrm{\mid }D{\eta }_{\epsilon }\mathrm{\mid }\to 0$ by F2. Thus the bulk integrals tend to ${\int }_{\mathrm{\Omega }}\mathrm{div}F$.

For every point of a face outside E, its positive distance from compact E exceeds epsilon eventually, so eta_epsilon vanishes there. Each face has finite area, and the flux is bounded by $\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty }}$. Since E is facewise null, F5 gives convergence of each face integral in step 1.1 to its full flux. There are finitely many faces, so steps 1.1–2.1 yield the claimed identity.

In a specified finite gluing, apply this formula to every piece. Shared faces have the same surface measure and opposite normals by F1, so their two integrands sum pointwise to zero off the null edge sets. Each regular face has n-dimensional volume zero: in graph coordinates Fubini gives zero from its singleton vertical sections, and a rigid coordinate change preserves volume. Thus adding the piece volumes counts the final domain integral exactly, and only exposed face fluxes remain. This proves the cancellation statement.