Euclidean divergence and the Stokes comparison

The Euclidean results here prove the divergence identity in every dimension $n\ge 2$ from graph integration, a finite partition, and explicit edge cutoffs. The earlier three-dimensional finite-gluing vector-calculus result is a narrower comparison. In the later smooth differential-form setting, the flux form ${\iota }_F(d{x}_1\wedge \cdots \wedge d{x}_n)$ has exterior derivative $(\mathrm{div}F)d{x}_1\wedge \cdots \wedge d{x}_n$, so manifold Stokes specializes to the smooth-boundary formula. That comparison supplies no dependency for this proof. Neither rough-boundary divergence nor Sobolev traces are asserted here.

Hunter §1.12, Theorem 1.46 and ensuing discussion, printed pp. 17–18 (PDF pp. 23–24). The manifold comparison is a forward explanatory comment, not a theorem used in this batch.