Surface integration on compact C1 hypersurfaces

Assume the ${\mathrm{A}\mathrm{C}}_{\omega }$ convention of Bounded C1 domains and their outward normals. Let S be a compact embedded $C^1$ hypersurface in ${\mathbb{R}}^n$, $n\ge 2$. Choose finitely many regular injective $C^1$ parametrizations $X_j:V_j\to S\cap O_j$ with $C^1$ coordinate transitions, and an ambient partition ${\chi }_j$ from Finite ambient partitions near compact sets, with supports compactly contained in the corresponding chart neighborhoods. Put $J_{X_j}=\sqrt{\det (D{X}_j^{T}D{X}_j)}$, using The Gram matrix $G(v_0,\dots ,v_{r-1})=(⟨v_i,v_j⟩{)}_{i,j<r}$ and Gram determinant, with empty value $1$.

For a nonnegative Borel f on S define ${\int }_{S}fdS={\sum }_j{\int }_{V_j}({\chi }_{j}f)(X_j(y))J_{X_j}(y)dy$, where the integrals on the right are those of The nonnegative Lebesgue integral and $0\cdot \mathrm{\infty }=0$. Set $S(A)={\int }_S{\mathbf{1}}_{A}dS$ for Borel A. For signed f with ${\int }_S\mathrm{\mid }f\mathrm{\mid }dS<\mathrm{\infty }$ use the difference of the positive and negative integrals. The same chart formula restricts to a compact Borel face contained in a regular patch. The empty surface has zero integral. Independence and finiteness are established by the following chart-independence lemma.

Hunter, §1.10.2, printed p. 15 (PDF p. 21), Gram surface density and partition patching.