stylistic:tone, structural:reformatting, quality:correction
For bounded open subsets of Euclidean space

We are going to prove the following:

Theorem — Let Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} be open and bounded with C 1 {\displaystyle C^{1}} boundary. If u {\displaystyle u} is C 1 {\displaystyle C^{1}} on an open neighborhood O {\displaystyle O} of Ω ¯ {\displaystyle {\overline {\Omega }}}, that is, u ∈ C 1 (O) {\displaystyle u\in C^{1}(O)}, then for each i ∈ { 1, …, n } {\displaystyle i\in \{1,\dots,n\}}, ∫ Ω u x i d V = ∫ ∂ Ω u ν i d S, {\displaystyle \int _{\Omega }u_{x_{i}}\,dV=\int _{\partial \Omega }u\nu _{i}\,dS,} where ν: ∂ Ω → R n {\displaystyle \nu :\partial \Omega \to \mathbb {R} ^{n}} is the outward pointing unit normal vector to ∂ Ω {\displaystyle \partial \Omega }.Equivalently, ∫ Ω ∇ u d V = ∫ ∂ Ω u ν d S. {\displaystyle \int _{\Omega }\nabla u\,dV=\int _{\partial \Omega }u\nu \,dS.}

Proof of Theorem.