Compact-support Stokes on Euclidean space

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 $-1$.

Form calculus extends locally across a manifold boundary: On smooth manifolds with boundary, the coordinate exterior derivative, pullback naturality, graded Leibniz rule, support containment, and Cartan identity hold for smooth forms: $d(F^{\ast }\alpha )=F^{\ast }(d\alpha ),d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1{)}^{\mathrm{deg}\alpha }\alpha \wedge d\beta ,$ $\mathrm{supp}d\alpha \subseteq \mathrm{supp}\alpha ,{\mathcal{L}}_X\alpha =d({\iota }_X\alpha )+{\iota }_{X}d\alpha \mathrm{.}$ For arbitrary smooth vector fields at boundary points, ${\mathcal{L}}_X$ is defined by local Euclidean extensions; a two-sided flow inside the manifold is not required.

A continuous function on a closed rectangle has repeated Riemann integrals in every coordinate order, all equal to its multiple integral: Let $Q={\prod }_{j<n}[a_j,b_j]\subseteq {\mathbb{R}}^n$, where $n\ge 1$ and every $a_j<b_j$. If $f:Q\to \mathbb{R}$ is continuous, then for every permutation of the coordinates the corresponding repeated Riemann integral exists and equals ${\int }_{Q}f$.

Newton–Leibniz needs only continuity on $[a,b]$, differentiability on $(a,b)$, and a Riemann-integrable extension of the interior derivative: Let $a<b$. Suppose $G:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$. If $f:[a,b]\to \mathbb{R}$ is Riemann integrable and $f(x)=G^{\mathrm{\prime }}(x)(a<x<b),$ then ${\int }_a^{b}f=G(b)-G(a)\mathrm{.}$ No derivative of $G$ at either endpoint is assumed, and the two endpoint values assigned to the integrable extension $f$ do not enter the conclusion.