面積分学習記録 (2)【ガウスの発散定理】

ある有界閉集合 $K\subset {\mathbb{R}}^N$ および $\psi \in C_0^{\mathrm{\infty }}\left({\mathbb{R}}^N\right)$ として，
$$\begin{array}{rl}& \begin{array}{r}\bar{\mathrm{\Omega }}\subset K\mathsf{\text{かつ}}\{\begin{array}{l}\psi (\bar{\mathrm{\Omega }})=\left\{1\right\}\\ \psi (K\setminus \bar{\mathrm{\Omega }})=[0,1)\\ \psi ({\mathbb{R}}^N\setminus K)=\left\{0\right\}\end{array}\end{array}\end{array}$$
を満たすものを取り，$u$ の ${\mathbb{R}}^N$ への拡張 ${\displaystyle \tilde{u}:={u|}_{{\mathbb{R}}^N}\mathbf{:}{\mathbb{R}}^N\to \mathbb{R}}$ を，
$$\begin{array}{rl}& \begin{array}{r}\tilde{u}\left(x\right):=\psi (x)\cdot u(x)=\{\begin{array}{cl}u(x)& (x\in \bar{\mathrm{\Omega }})\\ \psi (x)\cdot u(x)& (x\in K\setminus \bar{\mathrm{\Omega }})\\ 0& (x\in {\mathbb{R}}^N\setminus K)\end{array}\end{array}(\mathrm{\forall }x\in {\mathbb{R}}^N)\end{array}$$
として定める。このとき，
$$\begin{array}{rl}& \begin{array}{r}\tilde{u}\in C_0^1\left({\mathbb{R}}^N\right)\mathbf{.}\end{array}\end{array}$$
$$

$\partial \mathrm{\Omega }$ は有界閉集合かつ $C^1$ 級なので，
$$\begin{array}{rl}& \begin{array}{rl}& \mathrm{\exists }m\in \mathbb{Z}\cap [1,\mathrm{\infty })\mathbf{,}\mathrm{\exists }{\mathrm{\Gamma }}_1,\dots ,{\mathrm{\Gamma }}_m\subset \partial \mathrm{\Omega }\mathbf{:}\mathsf{\text{十分小さい、空でない、単連結な有界閉集合}}\mathbf{;}\\ & \mathrm{\forall }i\in \{1,\dots ,m\}\mathbf{,}\mathrm{\exists }{D}_i\subset {\mathbb{R}}^{N-1}\mathbf{,}\mathrm{\exists }{\gamma }_i\in C^1(D_i)\mathbf{,}\mathrm{\exists }{\sigma }_i\mathbf{:}{\mathbb{R}}^N\to {\mathbb{R}}^N\mathbf{:}\mathsf{\text{合同変換}}\mathbf{;}\\ & [{\sigma }_i\left({\mathrm{\Gamma }}_i\right)=G(D_i;{\gamma }_i)\mathsf{\text{かつ}}\bigcup _{i=1}^m{\mathrm{\Gamma }}_i=\partial \mathrm{\Omega }]\mathbf{.}\end{array}\end{array}$$
ここで，各 $i\in \{1,\dots ,m\}$ に対して，${\sigma }_i$ に対応する回転行列を，
$$\begin{array}{rl}& \begin{array}{r}{R}^i=[R_1^i\cdots R_N^i]=\left[\begin{array}{ccc}{R}_{1,1}^i& \cdots & R_{1,N}^i\\ ⋮& \ddots & ⋮\\ R_{N,1}^i& \cdots & R_{N,N}^i\end{array}\right]\in {\mathbb{R}}^{N\times N}\end{array}\end{array}$$
とする。
$$

関数 $\gamma \mathbf{:}{\mathbb{R}}^{N-1}\to \mathbb{R}$ を，
$$\begin{array}{rl}& \begin{array}{r}\gamma \left(x^{\prime }\right):=\{\begin{array}{cl}{\gamma }_1\left(x^{\prime }\right)& (x^{\prime }\in D_1)\\ ⋮& ⋮\\ {\gamma }_m\left(x^{\prime }\right)& (x^{\prime }\in D_m)\\ 0& {\displaystyle (x^{\prime }\in {\mathbb{R}}^{N-1}\setminus \bigcup _{i=1}^m{D}_i)}\end{array}\end{array}(\mathrm{\forall }{x}^{\prime }\in {\mathbb{R}}^{N-1})\end{array}$$
と定める。
また，各 $i\in \{1,\dots ,m\}$ に対して，法ベクトル場 ${\displaystyle {\mathit{\nu }}^i:=({\nu }_1^i,\dots ,{\nu }_N^i)\mathbf{:}{\sigma }_i\left({\mathrm{\Gamma }}_i\right)\to \mathbb{R}}$ を，
$$\begin{array}{rl}& \begin{array}{r}{\mathit{\nu }}^i\left(y\right)=\left[\begin{array}{c}{\nu }_1^i(y)\\ ⋮\\ {\nu }_N^i(y)\end{array}\right]:={\displaystyle \frac{1}{\sqrt{{|{\mathrm{\nabla }}_{y^{\prime }}\gamma (y^{\prime })|}^2+1}}}\left[\begin{array}{c}-{\mathrm{\nabla }}_{y^{\prime }}\gamma (y^{\prime })\\ 1\end{array}\right]\end{array}(\mathrm{\forall }y=(y^{\prime },y_N)\in {\sigma }_i\left({\mathrm{\Gamma }}_i\right))\end{array}$$
と定める。
$$
このとき，ある $i_0\in \{1,\dots ,m\}$ を選んだときに，
$$\begin{array}{rl}& \begin{array}{r}\mathrm{\forall }y\in {\sigma }_{i_0}\left({\mathrm{\Gamma }}_{i_0}\right)\mathbf{,}\mathrm{\exists }{x}^{i_0}\in {\mathrm{\Gamma }}_{i_0}\mathbf{;}{\left(R^{i_0}\right)}^{-1}{\mathit{\nu }}^{i_0}\left(y\right)=-\mathit{\nu }\left(x^{i_0}\right)\end{array}\end{array}$$
となる場合は，${\sigma }_{i_0}$ を取り直して，
$$\begin{array}{rl}& \begin{array}{r}\mathrm{\forall }y\in {\sigma }_{i_0}\left({\mathrm{\Gamma }}_{i_0}\right)\mathbf{,}\mathrm{\exists }{x}^{i_0}\in {\mathrm{\Gamma }}_{i_0}\mathbf{;}{\left(R^{i_0}\right)}^{-1}{\mathit{\nu }}^{i_0}\left(y\right)=\mathit{\nu }\left(x^{i_0}\right)\end{array}\end{array}$$
となるようにする。
$$
ここで，$R_{i_0}$ は正規行列であることから，
$$\begin{array}{rl}& \begin{array}{r}\mathrm{\forall }y\in {\sigma }_{i_0}\left({\mathrm{\Gamma }}_{i_0}\right)\mathbf{,}\mathrm{\exists }{x}^{i_0}\in {\mathrm{\Gamma }}_{i_0}\mathbf{;}{\left(R^{i_0}\right)}^{\mathsf{T}}{\mathit{\nu }}^{i_0}\left(y\right)={\left(R^{i_0}\right)}^{-1}{\mathit{\nu }}^{i_0}\left(y\right)=\mathit{\nu }\left(x^{i_0}\right)\mathbf{.}\end{array}\end{array}$$
$$

各 $i\in \{1,\dots ,m\}$ に対して，
$$\begin{array}{rl}& \begin{array}{r}{U}_i:=\{(x^{\prime },x_N)\in D_i\times \mathbb{R}|x_N\le \gamma (x^{\prime })\}\mathbf{,}{\bar{\mathrm{\Omega }}}_i:={\sigma }_i^{-1}(U_i\cap {\sigma }_i(\bar{\mathrm{\Omega }}))\end{array}\end{array}$$
と定める。
$$
${\displaystyle \bar{\mathrm{\Omega }}=\bigcup _{i=1}^m{\bar{\mathrm{\Omega }}}_i}$ であることを示す。
$$
まず，$x\in \bar{\mathrm{\Omega }}$ が与えられたとする。このとき，ある適当な $i_1\in \{1,\dots ,m\}$ を取れば，
$$\begin{array}{rl}& \{\begin{array}{rl}& {\sigma }_{i_1}\left(x\right)\in {\sigma }_{i_1}(\bar{\mathrm{\Omega }})\\ & \mathrm{\exists }{\tilde{x}}^{\prime }\in D_{i_1}\mathbf{;}\mathrm{\exists }{\tilde{x}}_N\in (-\mathrm{\infty },\gamma ({\tilde{x}}^{\prime })]\mathbf{;}{\sigma }_{i_1}\left(x\right)=({\tilde{x}}^{\prime },{\tilde{x}}_N)\in U_{i_1}\end{array}\end{array}$$
となるので，
$$\begin{array}{rl}& \begin{array}{rl}& {\sigma }_{i_1}\left(x\right)\in U_{i_1}\cap {\sigma }_{i_1}(\bar{\mathrm{\Omega }})\mathbf{.}\end{array}\end{array}$$
ゆえに，
$$\begin{array}{rl}& \begin{array}{rl}& x\in {\sigma }_{i_1}^{-1}(U_{i_1}\cap {\sigma }_{i_1}(\bar{\mathrm{\Omega }}))={\bar{\mathrm{\Omega }}}_{i_1}\subset \bigcup _{i=1}^m{\bar{\mathrm{\Omega }}}_i\end{array}\end{array}$$
となるので，
$$\begin{array}{rl}& \begin{array}{r}\bar{\mathrm{\Omega }}\subset \bigcup _{i=1}^m{\bar{\mathrm{\Omega }}}_i\mathbf{.}\end{array}\end{array}$$
次に，${\displaystyle y\in \bigcup _{i=1}^m{\bar{\mathrm{\Omega }}}_i}$ が与えられたとする。このとき，
$$\begin{array}{rl}& \begin{array}{rl}& \mathrm{\exists }{i}_2\in \{1,\dots ,m\}\mathbf{;}y\in {\bar{\mathrm{\Omega }}}_{i_2}\end{array}\end{array}$$
となるので，
$$\begin{array}{rl}& \begin{array}{rl}& {\sigma }_{i_2}\left(y\right)\in {\sigma }_{i_2}({\bar{\mathrm{\Omega }}}_{i_2})=U_{i_2}\cap {\sigma }_{i_2}(\bar{\mathrm{\Omega }})\subset {\sigma }_{i_2}(\bar{\mathrm{\Omega }})\mathbf{.}\end{array}\end{array}$$
ゆえに，
$$\begin{array}{rl}& \begin{array}{rl}& y\in {\sigma }_{i_2}^{-1}\left({\sigma }_{i_2}(\bar{\mathrm{\Omega }})\right)=\bar{\mathrm{\Omega }}\end{array}\end{array}$$
となるので，
$$\begin{array}{rl}& \begin{array}{r}\bar{\mathrm{\Omega }}\supset \bigcup _{i=1}^m{\bar{\mathrm{\Omega }}}_i\mathbf{.}\end{array}\end{array}$$
以上のことから，${\displaystyle \bar{\mathrm{\Omega }}=\bigcup _{i=1}^m{\bar{\mathrm{\Omega }}}_i}$ である。
$$

${\displaystyle {\mathrm{supp}}_{\bar{\mathrm{\Omega }}}\left(u\right)\subset \mathrm{\Omega }}$ の場合を考える。各 ${\displaystyle j\in \{1,\dots ,N\}}$ に対して，
$$\begin{array}{rl}& \begin{array}{rl}{\int }_{\mathrm{\Omega }}{\displaystyle \frac{\partial u}{\partial x_j}}dx& ={\int }_{\mathrm{\Omega }}{\displaystyle \frac{\partial \tilde{u}}{\partial x_j}}dx={\int }_{{\mathbb{R}}^N}{\displaystyle \frac{\partial \tilde{u}}{\partial x_j}}dx\\ & ={\int }_{{\mathbb{R}}^{N-1}}d\hat{x}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{\partial \tilde{u}}{\partial x_j}}\left(x\right)d{x}_j(\hat{x}:=(x_1,\dots ,x_{j-1},x_{j+1},\dots ,x_N))\\ & ={\int }_{{\mathbb{R}}^{N-1}}[\tilde{u}\left(x\right){]}_{x_j=-\mathrm{\infty }}^{x_j=\mathrm{\infty }}d\hat{x}\\ & ={\int }_{{\mathbb{R}}^{N-1}}0d\hat{x}=0\mathbf{,}\end{array}\\ & \begin{array}{rl}{\int }_{\partial \mathrm{\Omega }}u{\nu }_{j}dS& ={\int }_{\partial \mathrm{\Omega }}0\cdot {\nu }_{j}dS=0\end{array}\end{array}$$
となるので，
$$\begin{array}{rl}& \begin{array}{r}{\int }_{\mathrm{\Omega }}{\displaystyle \frac{\partial u}{\partial x_j}}dx=0={\int }_{\partial \mathrm{\Omega }}u{\nu }_{j}dS\mathbf{.}\end{array}\end{array}$$
$$

次に，${\displaystyle {\mathrm{supp}}_{\bar{\mathrm{\Omega }}}\left(u\right)\not\subset \mathrm{\Omega }}$ の場合を考える。
$$

ある有限な実数値関数列 ${\displaystyle {\left({\eta }_i\right)}_{i=1}^m\in C^{\mathrm{\infty }}({\mathbb{R}}^N{)}^{\{1,\dots ,m\}}}$ で，
$$\begin{array}{rl}& \begin{array}{r}\{\begin{array}{l}{\eta }_i\left({\mathbb{R}}^N\right)\subset [0,1]\\ \mathrm{supp}\left({\eta }_i\right)\subset {\bar{\mathrm{\Omega }}}_i\end{array}(\mathrm{\forall }i\in \{1,\dots ,m\})\mathsf{\text{かつ}}\sum _{i=1}^m{\eta }_i(x)=1(\mathrm{\forall }x\in \bar{\mathrm{\Omega }})\end{array}\end{array}$$
を満たすものを取る。このとき，各 ${\displaystyle i\in \{1,\dots ,m\}}$ に対して，
$$\begin{array}{rl}& \begin{array}{r}{\tilde{u}}_i:={\eta }_i\cdot \tilde{u}\end{array}\end{array}$$
と置くと，
$$\begin{array}{rl}& \begin{array}{r}{\tilde{u}}_i\in C_0^1\left({\mathbb{R}}^N\right)\mathsf{\text{かつ}}\mathrm{supp}\left({\tilde{u}}_i\right)\subset {\bar{\mathrm{\Omega }}}_i\mathbf{.}\end{array}\end{array}$$
$$
各 ${\displaystyle j\in \{1,\dots ,N\}}$ に対して，
$$\begin{array}{rl}& \begin{array}{rl}{\int }_{\mathrm{\Omega }}{\displaystyle \frac{\partial u}{\partial x_j}}dx& ={\int }_{\bar{\mathrm{\Omega }}}{\displaystyle \frac{\partial u}{\partial x_j}}dx={\int }_{\bar{\mathrm{\Omega }}}{\displaystyle \frac{\partial \tilde{u}}{\partial x_j}}dx\\ & ={\int }_{\bar{\mathrm{\Omega }}}\{0\cdot \tilde{u}\left(x\right)+1\cdot {\displaystyle \frac{\partial \tilde{u}}{\partial x_j}}\left(x\right)\}dx\\ & ={\int }_{\bar{\mathrm{\Omega }}}\{\sum _{i=1}^m{\displaystyle \frac{\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 {\displaystyle \frac{\partial \tilde{u}}{\partial x_j}}\left(x\right)\}dx\\ & ={\int }_{\bar{\mathrm{\Omega }}}\sum _{i=1}^m{\displaystyle \frac{\partial }{\partial x_j}}\{{\eta }_i(x)\cdot \tilde{u}\left(x\right)\}dx\\ & ={\int }_{\bar{\mathrm{\Omega }}}\sum _{i=1}^m{\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}\left(x\right)dx=\sum _{i=1}^m{\int }_{{\bar{\mathrm{\Omega }}}_i}{\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}\left(x\right)dx\\ & =\sum _{i=1}^m{\int }_{{\bar{\mathrm{\Omega }}}_i}{\displaystyle \frac{\partial ({\tilde{u}}_i\circ {\sigma }_i^{-1})}{\partial x_j}}\left({\sigma }_i\left(x\right)\right)dx\end{array}\end{array}$$
となるので，各 $i\in \{1,\dots ,m\}$ に対して，
$$\begin{array}{rl}& \begin{array}{r}{v}_i:={\tilde{u}}_i\circ {\sigma }_i^{-1}\mathbf{,}y:=(y_1,\dots ,y_N)={\sigma }_i\left(x\right)\end{array}(x\in {\bar{\mathrm{\Omega }}}_i)\end{array}$$
とすると，
$$\begin{array}{rl}& \begin{array}{rl}{\int }_{\mathrm{\Omega }}{\displaystyle \frac{\partial u}{\partial x_j}}dx& =\sum _{i=1}^m{\int }_{{\sigma }_i\left({\bar{\mathrm{\Omega }}}_i\right)}\sum _{k=1}^N{\displaystyle \frac{\partial v_i}{\partial y_k}}\cdot {\displaystyle \frac{\partial y_k}{\partial x_j}}\cdot |\det \left({\displaystyle \frac{\partial x}{\partial y}}\right)|dy\\ & =\sum _{i=1}^m{\int }_{{\sigma }_i\left({\bar{\mathrm{\Omega }}}_i\right)}\sum _{k=1}^N{\displaystyle \frac{\partial v_i}{\partial y_k}}\cdot R_{k,j}^i\cdot |\det \left({\left(R^i\right)}^{-1}\right)|dy\\ & =\sum _{i=1}^m\sum _{k=1}^N{\int }_{{\sigma }_i\left({\bar{\mathrm{\Omega }}}_i\right)}{\displaystyle \frac{\partial v_i}{\partial y_k}}\cdot R_{k,j}^i\cdot 1dy\\ & =\sum _{i=1}^m(\sum _{k=1}^{N-1}{\int }_{{\sigma }_i\left({\bar{\mathrm{\Omega }}}_i\right)}{\displaystyle \frac{\partial v_i}{\partial y_k}}\cdot R_{k,j}^{i}dy+{\int }_{{\sigma }_i\left({\bar{\mathrm{\Omega }}}_i\right)}{\displaystyle \frac{\partial v_i}{\partial y_N}}\cdot R_{N,j}^{i}dy)\mathbf{.}\end{array}\end{array}$$
$$

以下では，各 ${\displaystyle j,k\in \{1,\dots ,N\}}$ および ${\displaystyle i\in \{1,\dots ,m\}}$ を任意に固定し，
$$\begin{array}{rl}& \begin{array}{r}{I}_{k,j}^i:={\int }_{{\sigma }_i\left({\bar{\mathrm{\Omega }}}_i\right)}{\displaystyle \frac{\partial v_i}{\partial y_k}}\cdot R_{k,j}^{i}dy\end{array}\end{array}$$
と置く。ここで，${\displaystyle \mathrm{supp}\left({\tilde{u}}_i\right)\subset {\bar{\mathrm{\Omega }}}_i}$ であることに注意すると，
$$\begin{array}{rl}& \begin{array}{rl}{I}_{k,j}^i& ={\int }_{{\sigma }_i\left({\bar{\mathrm{\Omega }}}_i\right)}{\displaystyle \frac{\partial v_i}{\partial y_k}}\cdot R_{k,j}^{i}dy={\int }_{U_i}{\displaystyle \frac{\partial v_i}{\partial y_k}}\cdot R_{k,j}^{i}dy\\ & ={\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^{\gamma \left(y^{\prime }\right)}{\displaystyle \frac{\partial v_i}{\partial y_k}}(y^{\prime },y_N)\cdot R_{k,j}^{i}d{y}_N\mathbf{.}\end{array}\end{array}$$
$$
$k=N$ の場合は，
$$\begin{array}{rl}& \begin{array}{rl}{I}_{N,j}^i& ={\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^{\gamma \left(y^{\prime }\right)}{\displaystyle \frac{\partial v_i}{\partial y_N}}(y^{\prime },y_N)\cdot R_{N,j}^{i}d{y}_N\\ & ={\int }_{D_i}[v_i(y^{\prime },y_N){]}_{y_N=-\mathrm{\infty }}^{y_N=\gamma \left(y^{\prime }\right)}\cdot R_{N,j}^{i}d{y}^{\prime }\\ & ={\int }_{D_i}{v}_i(y^{\prime },\gamma \left(y^{\prime }\right))\cdot R_{N,j}^i\cdot 1d{y}^{\prime }\\ & ={\int }_{D_i}{v}_i(y^{\prime },\gamma \left(y^{\prime }\right))\cdot R_{N,j}^i\cdot {\nu }_N^i(y^{\prime },\gamma \left(y^{\prime }\right))\cdot \sqrt{{\left|{\mathrm{\nabla }}_{y^{\prime }}\gamma \left(y^{\prime }\right)\right|}^2+1}d{y}^{\prime }\\ & ={\int }_{{\sigma }_i\left({\mathrm{\Gamma }}_i\right)}{v}_i(y^{\prime },\gamma \left(y^{\prime }\right))\cdot [R_{1,j}^i\cdots R_{N,j}^i]\left[\begin{array}{c}\mathbf{0}\\ {\nu }_N^i(y^{\prime },\gamma \left(y^{\prime }\right))\end{array}\right]dS(y^{\prime },\gamma \left(y^{\prime }\right))\\ & ={\int }_{{\sigma }_i\left({\mathrm{\Gamma }}_i\right)}{v}_i\left(y\right)\cdot {\left(R_j^i\right)}^{\mathsf{T}}\left[\begin{array}{c}\mathbf{0}\\ {\nu }_N^i\left(y\right)\end{array}\right]dS\left(y\right)\mathbf{.}\end{array}\end{array}$$
$$
また，$k<N$ の場合は，$I_{k,j}^i$ において，
$$\begin{array}{rl}& \begin{array}{r}{y}_N=\gamma \left(y^{\prime }\right)+s\end{array}\end{array}$$
と変数変換すると，
$$\begin{array}{rl}& \begin{array}{r}d{y}_N=ds\mathbf{,}-\mathrm{\infty }<s\le 0\end{array}\end{array}$$
となり，
$$\begin{array}{rl}& \begin{array}{r}{w}_i(y^{\prime },s):=v_i(y^{\prime },\gamma \left(y^{\prime }\right)+s)\end{array}\end{array}$$
と置くと，
$$\begin{array}{rl}& \begin{array}{rl}{\displaystyle \frac{\partial w_i}{\partial y_k}}(y^{\prime },s)& ={\displaystyle \frac{\partial }{\partial y_k}}\left\{v_i(y^{\prime },\gamma \left(y^{\prime }\right)+s)\right\}\\ & =\sum _{p=1}^{N-1}{\displaystyle \frac{\partial v_i}{\partial y_p}}{\displaystyle \frac{\partial y_p}{\partial y_k}}+{\displaystyle \frac{\partial v_i}{\partial y_N}}{\displaystyle \frac{\partial y_N}{\partial y_k}}\\ & ={\displaystyle \frac{\partial v_i}{\partial y_k}}+{\displaystyle \frac{\partial v_i}{\partial y_N}}{\displaystyle \frac{\partial \gamma }{\partial y_k}}\left(y^{\prime }\right)\end{array}\end{array}$$
となるので，
$$\begin{array}{rl}& \begin{array}{rl}{I}_{k,j}^i& ={\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^{\gamma \left(y^{\prime }\right)}{\displaystyle \frac{\partial v_i}{\partial y_k}}(y^{\prime },y_N)\cdot R_{k,j}^{i}d{y}_N\\ & ={\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^0\{{\displaystyle \frac{\partial w_i}{\partial y_k}}(y^{\prime },s)-{\displaystyle \frac{\partial v_i}{\partial y_N}}{\displaystyle \frac{\partial \gamma }{\partial y_k}}\left(y^{\prime }\right)\}\cdot R_{k,j}^{i}ds\\ & ={\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^0{\displaystyle \frac{\partial w_i}{\partial y_k}}(y^{\prime },s)\cdot R_{k,j}^{i}ds+{\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^0{\displaystyle \frac{\partial v_i}{\partial y_N}}\{-{\displaystyle \frac{\partial \gamma }{\partial y_k}}\left(y^{\prime }\right)\}\cdot R_{k,j}^{i}ds\mathbf{.}\end{array}\end{array}$$
ゆえに，
$$\begin{array}{rl}& \begin{array}{rl}& J_1:={\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^0{\displaystyle \frac{\partial w_i}{\partial y_k}}(y^{\prime },s)\cdot R_{k,j}^{i}ds\mathbf{,}\\ & J_2:={\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^0{\displaystyle \frac{\partial v_i}{\partial y_N}}\{-{\displaystyle \frac{\partial \gamma }{\partial y_k}}\left(y^{\prime }\right)\}\cdot R_{k,j}^{i}ds\end{array}\end{array}$$
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，
$$\begin{array}{rl}& \begin{array}{rl}{J}_1& ={\iint }_{{\mathbb{R}}^N}{\displaystyle \frac{\partial w_i}{\partial y_k}}(y^{\prime },s)\cdot R_{k,j}^{i}d{y}^{\prime }ds\\ & ={\int }_{{\mathbb{R}}^N}{\displaystyle \frac{\partial w_i}{\partial y_k}}\left(y\right)\cdot R_{k,j}^{i}dy\\ & ={\int }_{{\mathbb{R}}^{N-1}}[w_i\left(y\right){]}_{y_k=-\mathrm{\infty }}^{y_k=-\mathrm{\infty }}\cdot R_{k,j}^{i}d\hat{y}(\hat{y}:=(y_1,\dots ,y_{k-1},y_{k+1},\dots ,y_N))\\ & ={\int }_{{\mathbb{R}}^{N-1}}0\cdot R_{k,j}^{i}d\hat{y}=0\mathbf{,}\end{array}\\ & \begin{array}{rl}{J}_2& ={\int }_{D_i}d{y}^{\prime }{\int }_{-\mathrm{\infty }}^{\gamma \left(y^{\prime }\right)}{\displaystyle \frac{\partial v_i}{\partial y_N}}(y^{\prime },y_N)\cdot \{-{\displaystyle \frac{\partial \gamma }{\partial y_k}}\left(y^{\prime }\right)\}\cdot R_{k,j}^{i}d{y}_N\\ & ={\int }_{D_i}{v}_i(y^{\prime },\gamma \left(y^{\prime }\right))\cdot \{-{\displaystyle \frac{\partial \gamma }{\partial y_k}}\left(y^{\prime }\right)\}\cdot R_{k,j}^{i}d{y}^{\prime }\\ & ={\int }_{D_i}{v}_i(y^{\prime },\gamma \left(y^{\prime }\right))\cdot R_{k,j}^i\cdot {\nu }_k^i(y^{\prime },\gamma \left(y^{\prime }\right))\cdot \sqrt{{\left|{\mathrm{\nabla }}_{y^{\prime }}\gamma \left(y^{\prime }\right)\right|}^2+1}d{y}^{\prime }\\ & ={\int }_{{\sigma }_i\left({\mathrm{\Gamma }}_i\right)}{v}_i(y^{\prime },\gamma \left(y^{\prime }\right))\cdot \left\{[R_{1,j}^i\cdots R_{N,j}^i]{\nu }_k^i(y^{\prime },\gamma \left(y^{\prime }\right))e_k\right\}dS(y^{\prime },\gamma \left(y^{\prime }\right))\\ & ={\int }_{{\sigma }_i\left({\mathrm{\Gamma }}_i\right)}{v}_i\left(y\right)\cdot \left\{{\left(R_j^i\right)}^{\mathsf{T}}{\nu }_k^i\left(y\right)e_k\right\}dS\left(y\right)\end{array}\end{array}$$
となるので，
$$\begin{array}{rl}& \begin{array}{r}{I}_{k,j}^i=J_1+J_2={\int }_{{\sigma }_i\left({\mathrm{\Gamma }}_i\right)}{v}_i\left(y\right)\cdot \left\{{\left(R_j^i\right)}^{\mathsf{T}}{\nu }_k^i\left(y\right)e_k\right\}dS\left(y\right)\mathbf{.}\end{array}\end{array}$$
$$