Computing form integrals by finite parametrizations

First record boundary control. Compactness and continuity give ${\bar{W}}_i=F_i({\bar{D}}_i)$ and ${\mathrm{\partial }}_M{W}_i\subset F_i(\mathrm{\partial }{D}_i)$: a limit of interior image points has a convergent parameter subsequence, and an interior parameter limit has image in $W_i$. Each $\mathrm{\partial }{D}_i$ is compact of content zero. Cover it by finitely many parameter neighborhoods with smooth coordinate extensions. Intersect smaller closed neighborhoods with $\mathrm{\partial }{D}_i$ and apply the $C^1$ null-image lemma on each extension domain. Thus $F_i(\mathrm{\partial }{D}_i)$ is content zero in every fixed relatively compact target chart, after finite localization. No derivative rank condition is used here.

By a finite chart partition of the compact support and linearity, it suffices to consider a form supported compactly inside a small chart $U$ whose coordinate domain $Y$ is a bounded rectangle or half-rectangle and whose chart extends past its artificial edges. Such charts come from restricting a larger chart; the Euclidean boundary of $Y$ has content zero. Put $C_i=\varphi (U\cap W_i)$. The boundaries of the bounded $C_i$ lie in $\mathrm{\partial }Y$ together with the chart images of ${\mathrm{\partial }}_M{W}_i$, so $C_i$ are Jordan measurable. The localized coefficient $f$ is bounded, zero near artificial edges, and Riemann integrable, including the genuine face.

For that localized coefficient, $f=0$ outside ${\bigcup }_i{\bar{C}}_i$. The disjoint $C_i$ overlap only on null boundaries after closure. Apply finite almost-partition additivity to the pieces $C_i$ and $Y\setminus {\bigcup }_i{C}_i$ (whose integral is zero since $f$ vanishes there except on those boundaries). Consequently the signed chart integral is ${\sigma }_{\varphi }{\sum }_i{\int }_{C_i}f$.

Fix $i$ and write $g=\varphi F_i$ on $A_i=F_i^{-1}(U\cap W_i)\subset D_i$. This is a diffeomorphism onto $C_i$. To justify substitution despite possible singularities at parameter boundary, let $h$ be the coefficient of $F_i^{\ast }\omega$ on $D_i$; it extends continuously to the compact ${\bar{D}}_i$ and is bounded, say by $B$. Choose a finite union $K\subset D_i$ of grid cubes, with disjoint interiors, covering all but a collar of $\mathrm{\partial }{D}_i$ of arbitrarily small volume. Images of that collar have arbitrarily small chart volume as well: finitely many smooth coordinate extensions have bounded derivatives and are Lipschitz on smaller convex neighborhoods; a cube of side $\delta$ maps into a cube of side at most $c\delta$, so total covering volume increases by at most a fixed factor. Such collars exist because $\mathrm{\partial }{D}_i$ has content zero.

On the compact part $K$, the nonzero support of $h$ lies in a compact subset of $A_i$, since the localized form is supported inside $U$. Subdivide or cover this compact part by finitely many cubes compactly contained in $A_i$, splitting overlaps along their faces. On each such compact Jordan piece, the published substitution theorem applies to $g$: it is injective, $C^1$, and has invertible derivative on the surrounding open subset of $A_i$. Pieces where the form vanishes contribute zero. Add these equalities. The omitted integrals on the parameter side are bounded by $B$ times collar volume; on the image side they are bounded by $\sup \mathrm{\mid }f\mathrm{\mid }$ times the image-collar covering volume. Let those bounds tend to zero. This proves ${\int }_{D_i}h={\sigma }_{\varphi }{\int }_{C_i}f$, with the sign supplied by orientation preservation.

Sum over $i$ and then over the finite chart localization. Empty support gives only zero coefficients; for $n=1$ the same collar estimate uses intervals and point boundaries. Degenerate Jacobians at boundary points are harmless because substitution was used only on compact subsets of the diffeomorphism domains.