Coordinate independence of chart integrals

Chart integral with its orientation sign: Let $M^n$ be oriented and $\omega$ a smooth top form with compact support contained in a connected chart $(U,\varphi )$. For $n\ge 1$ write $({\varphi }^{-1}{)}^{\ast }\omega =fd{x}^1\wedge \cdots \wedge d{x}^n\mathrm{.}$ Let ${\sigma }_{\varphi }\in \{1,-1\}$ be the sign of its coordinate frame relative to the chosen orientation. Define the chart integral by $I_{\varphi }(\omega )={\sigma }_{\varphi }{\int }_{{\mathbb{R}}^n}\tilde{f}(x)dx\mathrm{.}$ Here $\tilde{f}$ is the Riemann-integrable zero extension, including across a genuine half-space face, as in lem-chart-supported-coefficients-have-well-defined-riemann-integrable-half-space-extensions. For $n=0$, a connected chart is a point $p$, and set $I_p(\omega )=\epsilon (p)\omega (p)$ using its determinant-line sign. Empty support gives zero. Negative charts are allowed: the upper-half-line chart $u=b-t$ at the right endpoint of an increasing interval has sign $-1$.

Local side-preserving extensions of half-space transitions: Let $n\ge 1$ and $G:U\to V$ be a smooth diffeomorphism between relatively open subsets of $H^n$. At every $p\in U\cap \{x_n=0\}$ there are Euclidean open neighborhoods $O$ of $p$ and $O^{\mathrm{\prime }}$ of $G(p)$ and a smooth diffeomorphism $\widehat{G}:O\to O^{\mathrm{\prime }}$ extending $G$ locally, such that $\widehat{G}$ maps the positive, zero, and negative sides of $x_n=0$ onto the corresponding sides in $O^{\mathrm{\prime }}$.

A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage: Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, and let $g:U\to {\mathbb{R}}^n$ be injective and $C^1$, with $Dg(x)$ invertible on $U$. Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be compactly supported Riemann integrable and suppose $\mathrm{supp}f\subseteq g(U)$. Define $h(x)=\{\begin{array}{ll}f(g(x))\mathrm{\mid }\det Dg(x)\mathrm{\mid },& x\in U,\\ 0,& x\mathrm{\notin }U\mathrm{.}\end{array}$ Then $h$ is compactly supported Riemann integrable and ${\int }_{{\mathbb{R}}^n}f(y)dy={\int }_{{\mathbb{R}}^n}h(x)dx\mathrm{.}$

Pullback of forms is smooth functorial and preserves wedges: For a smooth map $F:M\to N$, pullback sends smooth differential forms on $N$ to smooth differential forms on $M$, is functorial, and satisfies $F^{\ast }(\alpha \wedge \beta )=F^{\ast }\alpha \wedge F^{\ast }\beta \mathrm{.}$

Smooth partitions of unity exist on manifolds with boundary: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Every open cover of a smooth manifold with boundary admits a smooth partition of unity subordinate to it.

Local finiteness near compact support: If $(C_i{)}_{i\in I}$ is a locally finite family of closed subsets of a manifold and $K$ is compact, only finitely many $C_i$ meet $K$. There is an open neighborhood of $K$ disjoint from all the other $C_i$. In particular, for a smooth partition of unity $({\rho }_i)$ and $\omega \in {\mathrm{\Omega }}_c^k(M)$, only finitely many ${\rho }_i\omega$ are nonzero.