Partition weights in two overlapping charts

Independence of atlas, partition and refinement: The compact-support integral on an oriented manifold is independent of the chart cover, coordinate maps, subordinate partition, and refinement. If $U\subset M$ is open and contains $\mathrm{supp}\omega$, with its restricted orientation, then ${\int }_U\omega {\mathrm{\mid }}_U={\int }_M\omega$.

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 $-1$.