Stokes for C1 one-forms on compact surface regions

By compactness and [F2], choose finitely many $C^2$ straightening neighborhoods of $D$ with compact coordinate closures, using the local models in the Statement at points on $\mathrm{\partial }M$. At an interior boundary arc with parametrization $\gamma (s)$, choose a transverse vector $v$ and choose the signs of $s$ and $t$ so the coordinate map $(s,t)\mapsto \gamma (s)+tv$ is orientation-preserving and $D$ lies in one half-plane; its derivative is invertible, so [F8] gives a $C^2$ inverse chart. At a corner in an interior chart, extend each incident arc across its endpoint by its second-order Taylor polynomial on the missing side; the value and the first two one-sided derivatives agree at the endpoint, so this is a $C^2$ extension. Order the two arc parameters so that $(s,t)\mapsto {\gamma }_1(s)+{\gamma }_2(t)-{\gamma }_1(0)$ is orientation-preserving. Its derivative is invertible by the linear independence of the one-sided tangent vectors, so [F8] gives a $C^2$ local diffeomorphism. After shrinking, the local model is a quadrant or its reentrant complement as in the Statement. Add $M\setminus D$ to this finite open cover and use [F3] for a smooth subordinate partition of unity. On $D$, only finitely many localized forms ${\eta }_i={\rho }_i\eta$ contribute, each with compact support in one straightening chart.

In an oriented coordinate plane, a compactly supported $C^1$ form $adx+bdy$ satisfies ${\int }_{{\mathbb{R}}^2}({\mathrm{\partial }}_{x}b-{\mathrm{\partial }}_{y}a)=0$, ${\int }_{y\ge 0}({\mathrm{\partial }}_{x}b-{\mathrm{\partial }}_{y}a)={\int }_{\mathbb{R}}a(x,0)dx$, and ${\int }_{x,y\ge 0}({\mathrm{\partial }}_{x}b-{\mathrm{\partial }}_{y}a)={\int }_0^{\mathrm{\infty }}a(x,0)dx-{\int }_0^{\mathrm{\infty }}b(0,y)dy$. For complex coefficients, apply the calculation to real and imaginary parts. Put the support in a rectangle; [F9] makes the coefficients and their first derivatives Riemann integrable, and [F10] makes them uniformly continuous. Double Riemann sums and iterated one-dimensional Riemann sums have the same limit, since their difference is bounded by the rectangle area times the modulus of continuity at the mesh diameter. Then [F7] evaluates the one-dimensional integrals of ${\mathrm{\partial }}_{x}b$ and ${\mathrm{\partial }}_{y}a$ at their endpoints, and the compact support removes the artificial outer edges. The half-plane and quadrant boundary signs are exactly the outward-normal-first convention [F6]. On the reentrant model, subtract the missing open quadrant from the plane; the two axis integrals have the opposite orientation, so the same identity holds.

A $C^2$ straightening map preserves the exterior derivative of a $C^1$ one-form by the chain rule: the terms involving its second derivatives cancel by symmetry. The compact-support change-of-variables formula [F4] carries each local area identity to $D$, while [F6] carries the axis orientations to the oriented boundary arcs. Summing over the partition from step 1.1 gives ${\int }_{D}d\eta ={\int }_{\mathrm{\partial }D}\eta$. The empty region and empty boundary give zero on both sides.