Finite continuous surface integrals on compact C1 boundaries

The boundary is closed and bounded, hence compact by F1. Consider the family of all available local one-sided graph neighborhoods and all smaller ambient chart neighborhoods whose compact closures lie inside such neighborhoods. This family covers $S$; compactness retains finitely many smaller neighborhoods and their corresponding charts, without selecting a chart separately for every point. The partition is constructed explicitly in the next two steps. Each graph chart is a homeomorphism onto its image, has rank two, and has $C^1$ transitions: on an overlap the inverse of the second graph chart is its rigid-coordinate projection onto the two base coordinates.

Consider all rational centers $a\in {\mathbb{Q}}^3$ and positive rational radii $r<R$ for which ${\bar{B}}_R(a)$ lies inside one original chart neighborhood $O_j$. Assign each pair to the least such index $j$. Their inner balls $B_r(a)$ cover $S$, since every point has a small ball inside one chart neighborhood, and rational centers and radii can be chosen with a strict margin around that point. F1 retains finitely many inner balls covering $S$. For each retained pair put $s_l=(r_l+R_l)\mathrm{/}2$ and $b_l(x)=\sigma \text{\hspace{0.17em}⁣}\left(\frac{s_l^2-\mathrm{\mid }x-a_l{\mathrm{\mid }}^2}{s_l^2-r_l^2}\right)\mathrm{.}$ By F2 each $b_l$ is smooth, between zero and one, equals one on ${\bar{B}}_{r_l}(a_l)$ and has support in ${\bar{B}}_{s_l}(a_l)⋐O_{j(l)}$. This is a finite explicit selection from one covering family, with no countable-choice premise.

For graph charts $X:U_X\to S\cap O_X$ and $Y:U_Y\to S\cap O_Y$, put $E=S\cap O_X\cap O_Y$, $V=X^{-1}(E)$ and $W=Y^{-1}(E)$. These parameter domains are open, and $T=Y^{-1}\circ X:V\to W$ is a $C^1$ diffeomorphism with $X=Y\circ T$. F4 gives $DX=(DY\circ T)DT$ and hence $J_X=(J_Y\circ T)\mathrm{\mid }\det DT\mathrm{\mid }$, with $J_X=\sqrt{\det (D{X}^{T}DX)}$; rank two makes both Gram densities positive. Let $w:S\to \mathbb{R}$ be continuous with compact support $C\subset E$. The function $p(z)=w(Y(z))J_Y(z)$ on $W$, extended by zero to ${\mathbb{R}}^2$, is continuous with support contained in the compact set $Y^{-1}(C)\subset W=T(V)$, so it is Riemann integrable by F5. Apply F3 with $n=2$ and this $T,p$. Its pulled-back integrand on $V$ is $w(X(y))J_X(y)$, so the two zero-extended coordinate integrals agree.

Put $s={\sum }_l{b}_l$ and $\theta =\sigma (4s-1)$. Then $s\ge 1$ on $S$ and $\theta =1$ on the neighborhood $\{s>1\mathrm{/}2\}$ of $S$. For $s>0$ define $c_l=\theta b_l\mathrm{/}s$, and extend it by zero on $s=0$. Since $\theta =0$ where $s\le 1\mathrm{/}4$, this extension is smooth. Each $c_l$ is nonnegative with compact support inside $O_{j(l)}$. Put ${\chi }_j={\sum }_{l:j(l)=j}{c}_l$, using zero for an empty index set. Then each support is a finite union of compact subsets of $O_j$, and ${\sum }_j{\chi }_j=\theta =1$ near $S$.

For each $j$, $S\cap \mathrm{supp}{\chi }_j$ is compact and contained in $S\cap O_j$. Its inverse under $X_j$ is compact inside $U_j$. Thus $g_j$ vanishes outside a compact subset of $U_j$, its extension by zero is continuous on ${\mathbb{R}}^2$, and F5 makes its integral finite and independent of the containing rectangle.

Given two chart partitions ${\chi }_j$ and ${\eta }_k$, insert ${\sum }_k{\eta }_k=1$ in each integral of the first sum. The support on $S$ of each ${\chi }_j{\eta }_k$ is compact inside both chart neighborhoods, so step 2.2 transfers this continuous overlap integrand to the second chart. Finite linearity in F5 and ${\sum }_j{\chi }_j=1$ recover exactly the second sum, proving chart and partition independence.

The graph derivative has columns $(e_i,{\mathrm{\partial }}_{i}h)$, so its Gram matrix has entries $G_{ij}={\delta }_{ij}+{\mathrm{\partial }}_{i}h{\mathrm{\partial }}_{j}h$ for $i,j\in \{1,2\}$, and $\det G=(1+({\mathrm{\partial }}_{1}h{)}^2)(1+({\mathrm{\partial }}_{2}h{)}^2)-({\mathrm{\partial }}_{1}h{)}^2({\mathrm{\partial }}_{2}h{)}^2=1+\mathrm{\mid }Dh{\mathrm{\mid }}^2$. The vector $(-Dh,1)$ is perpendicular to both tangent columns and points out of the subgraph. On overlaps both graph descriptions have the same tangent plane; of its two unit normals the exterior side selects the same one. For the upper-side region the outward normal is the negative of the displayed subgraph normal. Rigid coordinate maps transport the chosen exterior-side vector, including orientation-reversing maps, by their orthogonal linear part; the exterior-side criterion still gives agreement on overlaps. The graph formula is continuous. Componentwise finite linearity and the continuous dot product give all vector, tensor and flux assertions.

For the conditional agreement assertion only, assume Countable Choice. By step 4.1 each zero-extended $g_j$ is continuous and compactly supported in ${\mathbb{R}}^2$, hence bounded and Borel on a containing nondegenerate closed rectangle and Riemann integrable there. Apply F6 with $n=2$ to this rectangle. Its Riemann and Lebesgue integrals agree, and it vanishes outside the rectangle. Equality term by term in the finite chart sum gives equality of the continuous-field surface integrals; step 5.1 already makes the Riemann sum independent of the chosen charts and partition. This imports no Countable Choice premise into the preceding construction or independence proof, and asserts no extension to arbitrary measurable surface fields.