面積分学習記録 (2) 【ガウスの発散定理】

1. ある有界閉集合 $K \subset \mathbb{R}^{N}$ および $\psi \in C_{0}^{\infty} \left( \mathbb{R}^{N} \right)$ として，
   \begin{align*} \qquad & \begin{aligned} \overline{\Omega} \subset K \qquad \textsf{かつ} \qquad \left\{ \begin{array}{l} \psi ( \overline{\Omega} ) = \left\{ 1 \right\} \\[5pt] \psi ( K \setminus \overline{\Omega} ) = \left[ 0 ,\, 1 \right) \\[5pt] \psi ( \mathbb{R}^{N} \setminus K ) = \left\{ 0 \right\} \end{array} \right. \end{aligned} \end{align*}
   を満たすものを取り，$u$ の $\mathbb{R}^{N}$ への拡張 $ \displaystyle \tilde{u} := \left. u \right|_{\mathbb{R}^{N}} \, \boldsymbol{:} \, \mathbb{R}^{N} \to \mathbb{R} $ を，
   \begin{align*} \qquad & \begin{aligned} \tilde{u} \left( x \right) := \psi (x) \cdot u ( x ) = \left\{ \begin{array}{cl} u ( x ) &{\quad} \left( \ x \in \overline{\Omega} \ \right) \\[0pt] \psi (x) \cdot u ( x ) &{\quad} \left( \ x \in K \setminus \overline{\Omega} \ \right) \\[2.5pt] 0 &{\quad} \left( \ x \in \mathbb{R}^{N} \setminus K \ \right) \end{array} \right. \end{aligned} \qquad \left( \ \forall \, x \in \mathbb{R}^{N} \ \right) \end{align*}
   として定める。このとき，
   \begin{align*} \qquad & \begin{aligned} \tilde{u} \in C^{1}_{0} \left( \mathbb{R}^{N} \right) \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$

2. $\partial \Omega$ は有界閉集合かつ $C^{1}$ 級なので，
   \begin{align*} \qquad & \begin{aligned} & \exists \, m \in \mathbb{Z} \cap \left[ 1 ,\, \infty \right) \, \boldsymbol{,} \quad \exists \, \Gamma_{1} ,\, \dots ,\, \Gamma_{m} \subset \partial \Omega \, \boldsymbol{:} \, \textsf{十分小さい、空でない、単連結な有界閉集合} \ \boldsymbol{;} \\[7.5pt] & \forall \, i \in \left\{ \, 1 ,\, \dots ,\, m \, \right\} \, \boldsymbol{,} \quad \exists \, D_{i} \subset \mathbb{R}^{N-1} \, \boldsymbol{,} \quad \exists \, \gamma_{i} \in C^{1} ( D_{i} ) \, \boldsymbol{,} \quad \exists \, \sigma_{i} \, \boldsymbol{:} \, \mathbb{R}^{N} \to \mathbb{R}^{N} \, \boldsymbol{:} \, \textsf{合同変換} \ \boldsymbol{;} \\[7.5pt] & \left[ \ \sigma_{i} \left( \Gamma_{i} \right) = G \left( D_{i} \, ; \, \gamma_{i} \right) \qquad \textsf{かつ} \qquad \bigcup_{i=1}^{m} \Gamma_{i} = \partial \Omega \ \right] \, \boldsymbol{.} \end{aligned} \end{align*}
   ここで，各 $i \in \left\{ \, 1 ,\, \dots ,\, m \, \right\}$ に対して，$\sigma_{i}$ に対応する回転行列を，
   \begin{align*} \qquad & \begin{aligned} R^{i} = \Bigl[ \, R^{i}_{1} \ \ \cdots \ \ R^{i}_{N} \, \Bigr] = \left[ \begin{array}{ccc} R^{i}_{1 ,\, 1} & \cdots & R^{i}_{1 ,\, N} \\ \vdots & \ddots & \vdots \\ R^{i}_{N ,\, 1} & \cdots & R^{i}_{N ,\, N} \end{array} \right] \in \mathbb{R}^{N \times N} \end{aligned} \end{align*}
   とする。
   $\\[5pt]$

3. 関数 $\gamma \, \boldsymbol{:} \, \mathbb{R}^{N-1} \to \mathbb{R}$ を，
   \begin{align*} \qquad & \begin{aligned} \gamma \left( x' \right) := \left\{ \begin{array}{cl} \gamma_{1} \left( x' \right) &{\quad} \left( \ x' \in D_{1} \ \right) \\[0pt] \vdots &{\quad} {\hspace{20pt}} \vdots \\[0pt] \gamma_{m} \left( x' \right) &{\quad} \left( \ x' \in D_{m} \ \right) \\[5pt] 0 &{\quad} \displaystyle \left( \ x' \in \mathbb{R}^{N-1} \setminus \bigcup_{i=1}^{m} D_{i} \ \right) \end{array} \right. \end{aligned} \qquad \left( \ \forall \, x' \in \mathbb{R}^{N-1} \ \right) \end{align*}
   と定める。
   また，各 $i \in \left\{ \, 1 ,\, \dots ,\, m \, \right\}$ に対して，法ベクトル場 $ \displaystyle \boldsymbol{\nu}^{i} := \left( \nu_{1}^{i} ,\, \dots ,\, \nu_{N}^{i} \right) \, \boldsymbol{:} \, \sigma_{i} \left( \Gamma_{i} \right) \to \mathbb{R} $ を，
   \begin{align*} \qquad & \begin{aligned} \boldsymbol{\nu}^{i} \left( y \right) = \begin{bmatrix} \nu_{1}^{i} ( y ) \\[0pt] \vdots \\[0pt] \nu_{N}^{i} ( y ) \\[0pt] \end{bmatrix} := \dfrac{ 1 }{ \sqrt{ \left| \nabla_{\! \! y'} \, \gamma ( y' ) \right|^{2} + 1 } } \begin{bmatrix} - \nabla_{\! \! y'} \, \gamma ( y' ) \\[2.5pt] 1 \end{bmatrix} \end{aligned} \qquad \left( \ \forall \, y = \left( y' ,\, y_{N} \right) \in \sigma_{i} \left( \Gamma_{i} \right) \ \right) \end{align*}
   と定める。
   $\\[5pt]$
   このとき，ある $i_{0} \in \left\{ \, 1 ,\, \dots ,\, m \, \right\}$ を選んだときに，
   \begin{align*} \qquad & \begin{aligned} \forall \, y \in \sigma_{i_{0}} \left( \Gamma_{i_{0}} \right) \, \boldsymbol{,} \quad \exists \, x^{i_{0}} \in \Gamma_{i_{0}} \ \boldsymbol{;} \qquad \left( R^{i_{0}} \right)^{-1} \, \boldsymbol{\nu}^{i_{0}} \left( y \right) = - \boldsymbol{\nu} \left( x^{i_{0}} \right) \end{aligned} \end{align*}
   となる場合は，$\sigma_{i_{0}}$ を取り直して，
   \begin{align*} \qquad & \begin{aligned} \forall \, y \in \sigma_{i_{0}} \left( \Gamma_{i_{0}} \right) \, \boldsymbol{,} \quad \exists \, x^{i_{0}} \in \Gamma_{i_{0}} \ \boldsymbol{;} \qquad \left( R^{i_{0}} \right)^{-1} \, \boldsymbol{\nu}^{i_{0}} \left( y \right) = \boldsymbol{\nu} \left( x^{i_{0}} \right) \end{aligned} \end{align*}
   となるようにする。
   $\\[5pt]$
   ここで，$R_{i_{0}}$ は正規行列であることから，
   \begin{align*} \qquad & \begin{aligned} \forall \, y \in \sigma_{i_{0}} \left( \Gamma_{i_{0}} \right) \, \boldsymbol{,} \quad \exists \, x^{i_{0}} \in \Gamma_{i_{0}} \ \boldsymbol{;} \qquad \left( R^{i_{0}} \right)^{\mathsf{T}} \, \boldsymbol{\nu}^{i_{0}} \left( y \right) = \left( R^{i_{0}} \right)^{-1} \, \boldsymbol{\nu}^{i_{0}} \left( y \right) = \boldsymbol{\nu} \left( x^{i_{0}} \right) \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$

4. 各 $i \in \left\{ \, 1 ,\, \dots ,\, m \, \right\}$ に対して，
   \begin{align*} \qquad & \begin{aligned} U_{i} := \Bigl\{ \, \left( x' ,\, x_{N} \right) \in D_{i} \times \mathbb{R} \ \, \Big{|} \, \ x_{N} \leq \gamma (x') \, \Bigr\} \, \boldsymbol{,} \qquad \overline{ \Omega }_{i} := \sigma_{i}^{-1} \left( U_{i} \cap \sigma_{i} \bigl( \overline{ \Omega } \bigr) \right) \end{aligned} \end{align*}
   と定める。
   $\\[5pt]$
   $\displaystyle \overline{ \Omega } = \bigcup_{i=1}^{m} \overline{ \Omega }_{i} $ であることを示す。
   $\\[5pt]$
   まず，$x \in \overline{\Omega}$ が与えられたとする。このとき，ある適当な $i_{1} \in \left\{ \, 1 ,\, \dots ,\, m \, \right\}$ を取れば，
   \begin{align*} \qquad & \left\{ \begin{aligned} & \sigma_{i_{1}} \left( x \right) \in \sigma_{i_{1}} \bigl( \overline{ \Omega } \bigr) \\[10pt] & \exists \, \tilde{x}' \in D_{i_{1}} \ \boldsymbol{;} \quad \exists \, \tilde{x}_{N} \in \bigl( - \infty \, , \ \gamma ( \tilde{x}' ) \bigr] \ \boldsymbol{;} \qquad \sigma_{i_{1}} \left( x \right) = \left( \tilde{x}' ,\, \tilde{x}_{N} \right) \in U_{i_{1}} \end{aligned} \right. \end{align*}
   となるので，
   \begin{align*} \qquad & \begin{aligned} & \sigma_{i_{1}} \left( x \right) \in U_{i_{1}} \cap \sigma_{i_{1}} \bigl( \overline{\Omega} \bigr) \, \boldsymbol{.} \end{aligned} \end{align*}
   ゆえに，
   \begin{align*} \qquad & \begin{aligned} & x \in \sigma_{i_{1}}^{-1} \left( U_{i_{1}} \cap \sigma_{i_{1}} \bigl( \overline{\Omega} \bigr) \right) = \overline{ \Omega }_{i_{1}} \subset \bigcup_{i=1}^{m} \overline{ \Omega }_{i} \end{aligned} \end{align*}
   となるので，
   \begin{align*} \qquad & \begin{aligned} \overline{\Omega} \subset \bigcup_{i=1}^{m} \overline{ \Omega }_{i} \, \boldsymbol{.} \end{aligned} \end{align*}
   次に，$ \displaystyle y \in \bigcup_{i=1}^{m} \overline{ \Omega }_{i} $ が与えられたとする。このとき，
   \begin{align*} \qquad & \begin{aligned} & \exists \, i_{2} \in \left\{ \, 1 ,\, \dots ,\, m \, \right\} \ \boldsymbol{;} \qquad y \in \overline{ \Omega }_{i_{2}} \end{aligned} \end{align*}
   となるので，
   \begin{align*} \qquad & \begin{aligned} & \sigma_{i_{2}} \left( y \right) \in \sigma_{i_{2}} \bigl( \overline{ \Omega }_{i_{2}} \bigr) = U_{i_{2}} \cap \sigma_{i_{2}} \bigl( \overline{\Omega} \bigr) \subset \sigma_{i_{2}} \bigl( \overline{\Omega} \bigr) \, \boldsymbol{.} \end{aligned} \end{align*}
   ゆえに，
   \begin{align*} \qquad & \begin{aligned} & y \in \sigma_{i_{2}}^{-1} \left( \sigma_{i_{2}} \bigl( \overline{ \Omega } \bigr) \right) = \overline{ \Omega } \end{aligned} \end{align*}
   となるので，
   \begin{align*} \qquad & \begin{aligned} \overline{\Omega} \supset \bigcup_{i=1}^{m} \overline{ \Omega }_{i} \, \boldsymbol{.} \end{aligned} \end{align*}
   以上のことから，$ \displaystyle \overline{ \Omega } = \bigcup_{i=1}^{m} \overline{ \Omega }_{i} $ である。
   $\\[5pt]$

5. $ \displaystyle \operatorname{supp}_{\overline{\Omega}} \left( u \right) \subset \Omega $ の場合を考える。各 $ \displaystyle j \in \left\{ \, 1 ,\, \dots ,\, N \, \right\} $ に対して，
   \begin{align*} \qquad & \begin{aligned} \int_{ \Omega } \dfrac{\partial u}{\partial x_{j}} \, dx &= \int_{\Omega } \dfrac{\partial \tilde{u}}{\partial x_{j}} \, dx = \int_{\mathbb{R}^{N} } \dfrac{\partial \tilde{u}}{\partial x_{j}} \, dx \\[5pt] &= \int_{ \mathbb{R}^{N-1} } d \hat{x} \int_{-\infty}^{\infty} \dfrac{\partial \tilde{u}}{\partial x_{j}} \left( x \right) \, dx_{j} \qquad \left( \ \hat{x} := \left( x_{1} ,\, \dots ,\, x_{j-1} ,\, x_{j+1} ,\, \dots ,\, x_{N} \right) \ \right) \\[5pt] &= \int_{ \mathbb{R}^{N-1} } \Bigl[ \tilde{u} \left( x \right) \Bigr]_{x_{j} = -\infty}^{x_{j} = \infty} \, d \hat{x} \\[5pt] &= \int_{ \mathbb{R}^{N-1} } 0 \ d \hat{x} = 0 \, \boldsymbol{,} \end{aligned} \\[15pt] & \begin{aligned} \int_{ \partial \Omega } u \, \nu_{j} \ dS &= \int_{ \partial \Omega } 0 \cdot \nu_{j} \ dS = 0 \end{aligned} \end{align*}
   となるので，
   \begin{align*} \qquad & \begin{aligned} \int_{ \Omega } \dfrac{\partial u}{\partial x_{j}} \, dx = 0 = \int_{ \partial \Omega } u \, \nu_{j} \ dS \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$

6. 次に，$ \displaystyle \operatorname{supp}_{\overline{\Omega}} \left( u \right) \not\subset \Omega $ の場合を考える。
   $\\[5pt]$

   ある有限な実数値関数列 $ \displaystyle \left( \eta_{i} \right)_{i=1}^{m} \in C^{\infty} ( \mathbb{R}^{N} )^{ \left\{ \, 1 ,\, \dots ,\, m \, \right\} } $ で，
   \begin{align*} \qquad & \begin{aligned} \left\{ \begin{array}{l} \eta_{i} \left( \mathbb{R}^{N} \right) \subset \left[ 0 ,\, 1 \right] \\[5pt] \operatorname{supp} \left( \eta_{i} \right) \subset \overline{ \Omega }_{i} \end{array} \right. \quad \left( \, \forall \, i \in \left\{ \, 1 ,\, \dots ,\, m \, \right\} \, \right) \qquad \textsf{かつ} \qquad \sum_{i=1}^{m} \eta_{i} ( x ) = 1 \quad \left( \, \forall \, x \in \overline{\Omega} \, \right) \end{aligned} \end{align*}
   を満たすものを取る。このとき，各 $ \displaystyle i \in \left\{ \, 1 ,\, \dots ,\, m \, \right\} $ に対して，
   \begin{align*} \qquad & \begin{aligned} \tilde{u}_{i} := \eta_{i} \cdot \tilde{u} \end{aligned} \end{align*}
   と置くと，
   \begin{align*} \qquad & \begin{aligned} \tilde{u}_{i} \in C^{1}_{0} \left( \mathbb{R}^{N} \right) \qquad \textsf{かつ} \qquad \operatorname{supp} \left( \tilde{u}_{i} \right) \subset \overline{ \Omega }_{i} \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$
   各 $ \displaystyle j \in \left\{ \, 1 ,\, \dots ,\, N \, \right\} $ に対して，
   \begin{align*} \qquad & \begin{aligned} \int_{ \Omega } \dfrac{\partial u}{\partial x_{j}} \, dx &= \int_{ \overline{ \Omega } } \dfrac{\partial u}{\partial x_{j}} \, dx = \int_{ \overline{ \Omega } } \dfrac{\partial \tilde{u}}{\partial x_{j}} \, dx \\[5pt] &= \int_{ \overline{ \Omega } } \left\{ 0 \cdot \tilde{u} \left( x \right) + 1 \cdot \dfrac{\partial \tilde{u}}{\partial x_{j}} \left( x \right) \right\} \, dx \\[5pt] &= \int_{ \overline{ \Omega } } \left\{ \sum_{i=1}^{m} \dfrac{\partial \eta_{i}}{\partial x_{j}} \left( x \right) \cdot \tilde{u} \left( x \right) + \sum_{i=1}^{m} \eta_{i} \left( x \right) \cdot \dfrac{\partial \tilde{u}}{\partial x_{j}} \left( x \right) \right\} \, dx \\[5pt] &= \int_{ \overline{ \Omega } } \sum_{i=1}^{m} \dfrac{\partial}{\partial x_{j}} \left\{ \eta_{i} ( x ) \cdot \tilde{u} \left( x \right) \right\} \, dx \\[5pt] &= \int_{ \overline{ \Omega } } \sum_{i=1}^{m} \dfrac{\partial \tilde{u}_{i}}{\partial x_{j}} \left( x \right) \, dx = \sum_{i=1}^{m} \int_{ \overline{ \Omega }_{i} } \dfrac{\partial \tilde{u}_{i}}{\partial x_{j}} \left( x \right) \, dx \\[5pt] &= \sum_{i=1}^{m} \int_{ \overline{ \Omega }_{i} } \dfrac{ \partial \left( \tilde{u}_{i} \circ \sigma_{i}^{-1} \right) }{\partial x_{j}} \left( \sigma_{i} \left( x \right) \right) \, dx \end{aligned} \end{align*}
   となるので，各 $i \in \left\{ 1 ,\, \dots ,\, m \right\}$ に対して，
   \begin{align*} \qquad & \begin{aligned} v_{i} := \tilde{u}_{i} \circ \sigma_{i}^{-1} \, \boldsymbol{,} \qquad y := \left( y_{1} ,\, \dots ,\, y_{N} \right) = \sigma_{i} \left( x \right) \end{aligned} \quad \left( \, x \in \overline{ \Omega }_{i} \, \right) \end{align*}
   とすると，
   \begin{align*} \qquad & \begin{aligned} \int_{ \Omega } \dfrac{\partial u}{\partial x_{j}} \, dx &= \sum_{i=1}^{m} \int_{ \sigma_{i} \left( \overline{ \Omega }_{i} \right) } \sum_{k=1}^{N} \dfrac{\partial v_{i}}{\partial y_{k}} \cdot \dfrac{\partial y_{k}}{\partial x_{j}} \cdot \left| \det \left( \dfrac{\partial x}{\partial y} \right) \right| \, dy \\[5pt] &= \sum_{i=1}^{m} \int_{ \sigma_{i} \left( \overline{ \Omega }_{i} \right) } \sum_{k=1}^{N} \dfrac{\partial v_{i}}{\partial y_{k}} \cdot R_{k ,\, j}^{i} \cdot \left| \det \left( \left( R^{i} \right)^{-1} \right) \right| \, dy \\[5pt] &= \sum_{i=1}^{m} \sum_{k=1}^{N} \int_{ \sigma_{i} \left( \overline{ \Omega }_{i} \right) } \dfrac{\partial v_{i}}{\partial y_{k}} \cdot R_{k ,\, j}^{i} \cdot 1 \, dy \\[5pt] &= \sum_{i=1}^{m} \left( \sum_{k=1}^{N-1} \int_{ \sigma_{i} \left( \overline{ \Omega }_{i} \right) } \dfrac{\partial v_{i}}{\partial y_{k}} \cdot R_{k ,\, j}^{i} \, dy + \int_{ \sigma_{i} \left( \overline{ \Omega }_{i} \right) } \dfrac{\partial v_{i}}{\partial y_{N}} \cdot R_{N ,\, j}^{i} \, dy \right) \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$

7. 以下では，各 $ \displaystyle j ,\, k \in \left\{ \, 1 ,\, \dots ,\, N \, \right\} $ および $ \displaystyle i \in \left\{ \, 1 ,\, \dots ,\, m \, \right\} $ を任意に固定し，
   \begin{align*} \qquad & \begin{aligned} I_{k ,\, j}^{i} := \int_{ \sigma_{i} \left( \overline{ \Omega }_{i} \right) } \dfrac{\partial v_{i}}{\partial y_{k}} \cdot R_{k ,\, j}^{i} \, dy \end{aligned} \end{align*}
   と置く。ここで，$ \displaystyle \operatorname{supp} \left( \tilde{u}_{i} \right) \subset \overline{ \Omega }_{i} $ であることに注意すると，
   \begin{align*} \qquad & \begin{aligned} I_{k ,\, j}^{i} &= \int_{ \sigma_{i} \left( \overline{ \Omega }_{i} \right) } \dfrac{\partial v_{i}}{\partial y_{k}} \cdot R_{k ,\, j}^{i} \, dy = \int_{ U_{i} } \dfrac{\partial v_{i}}{\partial y_{k}} \cdot R_{k ,\, j}^{i} \, dy \\[5pt] &= \int_{D_{i}} dy' \int_{ -\infty }^{ \gamma \left( y' \right) } \dfrac{\partial v_{i}}{\partial y_{k}} \left( y' ,\, y_{N} \right) \cdot R_{k ,\, j}^{i} \, dy_{N} \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$
   $k = N$ の場合は，
   \begin{align*} \qquad & \begin{aligned} I_{N ,\, j}^{i} &= \int_{D_{i}} dy' \int_{ -\infty }^{ \gamma \left( y' \right) } \dfrac{\partial v_{i}}{\partial y_{N}} \left( y' ,\, y_{N} \right) \cdot R_{N ,\, j}^{i} \, dy_{N} \\[5pt] &= \int_{D_{i}} \Bigl[ v_{i} \left( y' ,\, y_{N} \right) \Bigr]_{ y_{N} = -\infty }^{ y_{N} = \gamma \left( y' \right) } \cdot R_{N ,\, j}^{i} \, dy' \\[5pt] &= \int_{D_{i}} v_{i} \left( y' ,\, \gamma \left( y' \right) \right) \cdot R_{N ,\, j}^{i} \cdot 1 \, dy' \\[5pt] &= \int_{D_{i}} v_{i} \left( y' ,\, \gamma \left( y' \right) \right) \cdot R_{N ,\, j}^{i} \cdot \nu_{N}^{i} \left( y' ,\, \gamma \left( y' \right) \right) \cdot \sqrt{ \left| \nabla_{\! \! y'} \, \gamma \left( y' \right) \right|^{2} + 1 } \, dy' \\[5pt] &= \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y' ,\, \gamma \left( y' \right) \right) \cdot \left[ \, R_{1 ,\, j}^{i} \ \ \cdots \ \ R_{N ,\, j}^{i} \, \right] \left[ \, \begin{array}{c} \boldsymbol{0} \\ \nu_{N}^{i} \left( y' ,\, \gamma \left( y' \right) \right) \end{array} \, \right] \, dS \left( y' ,\, \gamma \left( y' \right) \right) \\[5pt] &= \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y \right) \cdot \left( \, R_{j}^{i} \, \right)^{ \mathsf{T} } \left[ \, \begin{array}{c} \boldsymbol{0} \\ \nu_{N}^{i} \left( y \right) \end{array} \, \right] \, dS \left( y \right) \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$
   また，$k < N$ の場合は，$I_{k ,\, j}^{i}$ において，
   \begin{align*} \qquad & \begin{aligned} y_{N} = \gamma \left( y' \right) + s \end{aligned} \end{align*}
   と変数変換すると，
   \begin{align*} \qquad & \begin{aligned} dy_{N} = ds \, \boldsymbol{,} \qquad - \infty < s \leq 0 \end{aligned} \end{align*}
   となり，
   \begin{align*} \qquad & \begin{aligned} w_{i} \left( y' ,\, s \right) := v_{i} \left( y' ,\, \gamma \left( y' \right) + s \right) \end{aligned} \end{align*}
   と置くと，
   \begin{align*} \qquad & \begin{aligned} \dfrac{\partial w_{i}}{\partial y_{k}} \left( y' ,\, s \right) &= \dfrac{\partial}{\partial y_{k}} \left\{ v_{i} \left( y' ,\, \gamma \left( y' \right) + s \right) \right\} \\[5pt] &= \sum_{p=1}^{N-1} \dfrac{\partial v_{i}}{\partial y_{p}} \dfrac{\partial y_{p}}{\partial y_{k}} + \dfrac{\partial v_{i}}{\partial y_{N}} \dfrac{\partial y_{N}}{\partial y_{k}} \\[5pt] &= \dfrac{\partial v_{i}}{\partial y_{k}} + \dfrac{\partial v_{i}}{\partial y_{N}} \dfrac{\partial \gamma}{\partial y_{k}} \left( y' \right) \end{aligned} \end{align*}
   となるので，
   \begin{align*} \qquad & \begin{aligned} I_{k ,\, j}^{i} &= \int_{D_{i}} dy' \int_{ -\infty }^{ \gamma \left( y' \right) } \dfrac{\partial v_{i}}{\partial y_{k}} \left( y' ,\, y_{N} \right) \cdot R_{k ,\, j}^{i} \, dy_{N} \\[5pt] &= \int_{D_{i}} dy' \int_{ -\infty }^{ 0 } \left\{ \dfrac{\partial w_{i}}{\partial y_{k}} \left( y' ,\, s \right) - \dfrac{\partial v_{i}}{\partial y_{N}} \dfrac{\partial \gamma}{\partial y_{k}} \left( y' \right) \right\} \cdot R_{k ,\, j}^{i} \, ds \\[5pt] &= \int_{D_{i}} dy' \int_{ -\infty }^{ 0 } \dfrac{\partial w_{i}}{\partial y_{k}} \left( y' ,\, s \right) \cdot R_{k ,\, j}^{i} \, ds + \int_{D_{i}} dy' \int_{ -\infty }^{ 0 } \dfrac{\partial v_{i}}{\partial y_{N}} \left\{ - \dfrac{\partial \gamma}{\partial y_{k}} \left( y' \right) \right\} \cdot R_{k ,\, j}^{i} \, ds \, \boldsymbol{.} \end{aligned} \end{align*}
   ゆえに，
   \begin{align*} \qquad & \begin{aligned} & J_{1} := \int_{D_{i}} dy' \int_{ -\infty }^{ 0 } \dfrac{\partial w_{i}}{\partial y_{k}} \left( y' ,\, s \right) \cdot R_{k ,\, j}^{i} \, ds \, \boldsymbol{,} \\[10pt] & J_{2} := \int_{D_{i}} dy' \int_{ -\infty }^{ 0 } \dfrac{\partial v_{i}}{\partial y_{N}} \left\{ - \dfrac{\partial \gamma}{\partial y_{k}} \left( y' \right) \right\} \cdot R_{k ,\, j}^{i} \, ds \end{aligned} \end{align*}
   と置き，$ \displaystyle e_{1} ,\, \dots ,\, e_{N} $ を $\mathbb{R}^{N}$ 内における基本ベクトルとすると，
   \begin{align*} \qquad & \begin{aligned} J_{1} &= \iint_{ \mathbb{R}^{N} } \dfrac{\partial w_{i}}{\partial y_{k}} \left( y' ,\, s \right) \cdot R_{k ,\, j}^{i} \, dy' \, ds \\[5pt] &= \int_{ \mathbb{R}^{N} } \dfrac{\partial w_{i}}{\partial y_{k}} \left( y \right) \cdot R_{k ,\, j}^{i} \, dy \\[5pt] &= \int_{ \mathbb{R}^{N-1} } \Bigl[ w_{i} \left( y \right) \Bigr]_{ y_{k} = - \infty }^{ y_{k} = - \infty } \cdot R_{k ,\, j}^{i} \, d \hat{y} \qquad \left( \ \hat{y} := \left( y_{1} ,\, \dots ,\, y_{k-1} ,\, y_{k+1} ,\, \dots ,\, y_{N} \right) \ \right) \\[5pt] &= \int_{ \mathbb{R}^{N-1} } 0 \cdot R_{k ,\, j}^{i} \, d \hat{y} = 0 \, \boldsymbol{,} \end{aligned} \\[15pt] & \begin{aligned} J_{2} &= \int_{D_{i}} dy' \int_{ -\infty }^{ \gamma \left( y' \right) } \dfrac{\partial v_{i}}{\partial y_{N}} \left( y' ,\, y_{N} \right) \cdot \left\{ - \dfrac{\partial \gamma}{\partial y_{k}} \left( y' \right) \right\} \cdot R_{k ,\, j}^{i} \, dy_{N} \\[5pt] &= \int_{D_{i}} v_{i} \left( y' ,\, \gamma \left( y' \right) \right) \cdot \left\{ - \dfrac{\partial \gamma}{\partial y_{k}} \left( y' \right) \right\} \cdot R_{k ,\, j}^{i} \, dy' \\[5pt] &= \int_{D_{i}} v_{i} \left( y' ,\, \gamma \left( y' \right) \right) \cdot R_{k ,\, j}^{i} \cdot \nu_{k}^{i} \left( y' ,\, \gamma \left( y' \right) \right) \cdot \sqrt{ \left| \nabla_{\! \! y'} \, \gamma \left( y' \right) \right|^{2} + 1 } \, dy' \\[5pt] &= \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y' ,\, \gamma \left( y' \right) \right) \cdot \left\{ \left[ \, R_{1 ,\, j}^{i} \ \ \cdots \ \ R_{N ,\, j}^{i} \, \right] \, \nu_{k}^{i} \left( y' ,\, \gamma \left( y' \right) \right) e_{k} \right\} \, dS \left( y' ,\, \gamma \left( y' \right) \right) \\[5pt] &= \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y \right) \cdot \left\{ \left( \, R_{j}^{i} \, \right)^{ \mathsf{T} } \, \nu_{k}^{i} \left( y \right) e_{k} \right\} \, dS \left( y \right) \end{aligned} \end{align*}
   となるので，
   \begin{align*} \qquad & \begin{aligned} I_{k ,\, j}^{i} = J_{1} + J_{2} = \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y \right) \cdot \left\{ \left( \, R_{j}^{i} \, \right)^{ \mathsf{T} } \, \nu_{k}^{i} \left( y \right) e_{k} \right\} \, dS \left( y \right) \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$

8. よって，
   \begin{align*} \qquad & \begin{aligned} \sum_{k=1}^{N} I_{k ,\, j}^{i} &= \sum_{k=1}^{N-1} I_{k ,\, j}^{i} + I_{N ,\, j}^{i} \\[5pt] &= \sum_{k=1}^{N-1} \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y \right) \cdot \left\{ \left( \, R_{j}^{i} \, \right)^{ \mathsf{T} } \, \nu_{k}^{i} \left( y \right) e_{k} \right\} \, dS \left( y \right) \\ &{\hspace{50pt}}+ \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y \right) \cdot \left\{ \left( \, R_{j}^{i} \, \right)^{ \mathsf{T} } \, \nu_{N}^{i} \left( y \right) e_{N} \right\} \, dS \left( y \right) \\[5pt] &= \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y \right) \cdot \left\{ \left( \, R_{j}^{i} \, \right)^{ \mathsf{T} } \, \boldsymbol{\nu}^{i} \left( y \right) \right\} \, dS \left( y \right) \end{aligned} \end{align*}
   となるので，
   \begin{align*} \qquad & \begin{aligned} \int_{ \Omega } \dfrac{\partial u}{\partial x_{j}} \, dx &= \sum_{i=1}^{m} \left( \sum_{k=1}^{N} I_{k ,\, j}^{i} \right) = \sum_{i=1}^{m} \int_{ \sigma_{i} \left( \Gamma_{i} \right) } v_{i} \left( y \right) \cdot \left\{ \left( \, R_{j}^{i} \, \right)^{ \mathsf{T} } \, \boldsymbol{\nu}^{i} \left( y \right) \right\} \, dS \left( y \right) \\[5pt] &= \sum_{i=1}^{m} \int_{ \sigma_{i} \left( \Gamma_{i} \right) } \tilde{u}_{i} \left( \sigma_{i}^{-1} \left( y \right) \right) \cdot \left\{ \left( \, R_{j}^{i} \, \right)^{ \mathsf{T} } \, \boldsymbol{\nu}^{i} \left( y \right) \right\} \, dS \left( y \right) \\[5pt] &= \sum_{i=1}^{m} \int_{ \Gamma_{i} } \tilde{u}_{i} \left( x \right) \cdot \nu_{j} \left( x \right) \, dS \left( x \right) \qquad \left( \, \textsf{面積分の定義 および } \left( R^{i} \right)^{\mathsf{T}} \, \boldsymbol{\nu}^{i} \left( y \right) = \boldsymbol{\nu} \left( x \right) \textsf{ より} \, \right) \\[5pt] &= \int_{ \partial \Omega } \sum_{i=1}^{m} \tilde{u}_{i} \left( x \right) \cdot \nu_{j} \left( x \right) \, dS \left( x \right) \\[5pt] &= \int_{ \partial \Omega } \sum_{i=1}^{m} \eta_{i} \left( x \right) \cdot \left\{ \tilde{u} \left( x \right) \cdot \nu_{j} \left( x \right) \right\} \, dS \left( x \right) = \int_{ \partial \Omega } \tilde{u} \cdot \nu_{j} \, dS \\[5pt] &= \int_{ \partial \Omega } u \cdot \nu_{j} \, dS \, \boldsymbol{.} \end{aligned} \end{align*}
   $\\[5pt]$

9. したがって，以上のことから，
   \begin{align*} \qquad & \begin{aligned} \int_{ \Omega } \dfrac{\partial u}{\partial x_{j}} \, dx &= \int_{ \partial \Omega } u \cdot \nu_{j} \, dS \end{aligned} \end{align*}
   が成り立つ。$\hspace{0pt} \blacksquare$

$\displaystyle \boldsymbol{u} := \left( u_{1} ,\, \dots ,\, u_{N} \right) \, \boldsymbol{,} \ \ \boldsymbol{\nu} := \left( \nu_{1} ,\, \dots ,\, \nu_{N} \right) $ とする。このとき，ガウス--グリーンの定理から，
\begin{align*} \qquad & \begin{aligned} \int_{ \Omega } \nabla \boldsymbol{\cdot} \boldsymbol{u} \, dx &= \int_{ \Omega } \sum_{k=1}^{N} \dfrac{\partial u_{k}}{\partial x_{k}} \, dx = \sum_{k=1}^{N} \int_{ \Omega } \dfrac{\partial u_{k}}{\partial x_{k}} \, dx = \sum_{k=1}^{N} \int_{ \partial \Omega } u_{k} \cdot \nu_{k} \, dS \\[5pt] &= \int_{ \partial \Omega } \sum_{k=1}^{N} u_{k} \cdot \nu_{k} \, dS = \int_{ \partial \Omega } \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nu} \, dS \, \boldsymbol{.} \end{aligned} \end{align*}