内部座標に対する微分の式

ここでは、内部座標の定義とその微分の式を記す。

数式の notation

・ベクトルの下付き添字の2つのアルファベットは $\mathbf {r} _{ij}=\mathbf {r} _{i}-\mathbf {r} _{j}$ などのような意味となる。

・単位ベクトルとして $\mathbf {e} _{ij}$ などの表記を用いる。

・成分の表記として ${x}_{ij}$ or $(\mathbf {e} _{ij})_{x}$ などの表記を用いる。

結合長

・定義

$R=|\mathbf {r} _{ij}|$

・微分

${\begin{aligned}{\frac {\partial R}{\partial \mathbf {r} _{i}}}&={\frac {\mathbf {r} _{ij}}{|\mathbf {r} _{ij}|}}=\mathbf {e} _{ij}\\{\frac {\partial R}{\partial \mathbf {r} _{j}}}&=-{\frac {\mathbf {r} _{ij}}{|\mathbf {r} _{ij}|}}=-\mathbf {e} _{ij}\\\end{aligned}}$

・微分の導出

○ $\mathbf {r} _{i}$ での微分

${\begin{aligned}{\frac {\partial R}{\partial x_{i}}}&={\frac {\partial |\mathbf {r} _{ij}|}{\partial x_{i}}}\\&={\frac {\partial {\sqrt {x_{ij}^{2}+y_{ij}^{2}+z_{ij}^{2}}}}{\partial x_{i}}}\\&={\frac {2x_{ij}}{2{\sqrt {x_{ij}^{2}+y_{ij}^{2}+z_{ij}^{2}}}}}\\&={\frac {x_{ij}}{|\mathbf {r} _{ij}|}}\\\end{aligned}}$

よって、

${\begin{aligned}{\frac {\partial R}{\partial \mathbf {r} _{i}}}&={\frac {\mathbf {r} _{ij}}{|\mathbf {r} _{ij}|}}\\&=\mathbf {e} _{ij}\\\end{aligned}}$

○ $\mathbf {r} _{j}$ での微分

${\begin{aligned}{\frac {\partial R}{\partial \mathbf {r} _{j}}}&=-{\frac {\partial R}{\partial \mathbf {r} _{i}}}=-{\frac {\mathbf {r} _{ij}}{|\mathbf {r} _{ij}|}}\\&=-\mathbf {e} _{ij}\\\end{aligned}}$

結合角

・定義

$\theta =\arccos(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})=\arccos \left({\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{ij}||\mathbf {r} _{kj}|}}\right)$

・微分

${\begin{aligned}{\frac {\partial \theta }{\partial \mathbf {r} _{i}}}&=-{\frac {(\mathbf {e} _{z}\times \mathbf {e} _{ij})}{|\mathbf {r} _{ij}|}}\\&=-{\frac {\mathbf {e} _{kj}-(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})\mathbf {e} _{ij}}{|\mathbf {r} _{ij}|\sin \theta }}\\{\frac {\partial \theta }{\partial \mathbf {r} _{j}}}&=-{\frac {\partial \theta }{\partial \mathbf {r} _{i}}}-{\frac {\partial \theta }{\partial \mathbf {r} _{k}}}\\&={\frac {(\mathbf {e} _{z}\times \mathbf {e} _{ij})}{|\mathbf {r} _{ij}|}}+{\frac {(\mathbf {e} _{z}\times \mathbf {e} _{kj})}{|\mathbf {r} _{kj}|}}\\&={\frac {\mathbf {e} _{kj}-(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})\mathbf {e} _{ij}}{|\mathbf {r} _{ij}|\sin \theta }}+{\frac {\mathbf {e} _{ij}-(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})\mathbf {e} _{kj}}{|\mathbf {r} _{kj}|\sin \theta }}\\{\frac {\partial \theta }{\partial \mathbf {r} _{k}}}&=-{\frac {(\mathbf {e} _{z}\times \mathbf {e} _{kj})}{|\mathbf {r} _{kj}|}}\\&=-{\frac {\mathbf {e} _{ij}-(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})\mathbf {e} _{kj}}{|\mathbf {r} _{kj}|\sin \theta }}\\\end{aligned}}$

ここで、

$\mathbf {e} _{z}={\frac {\mathbf {r} _{ij}\times \mathbf {r} _{kj}}{|\mathbf {r} _{ij}\times \mathbf {r} _{kj}|}}$

(1)

$\mathbf {e} _{x}=\mathbf {e} _{ij}={\frac {\mathbf {r} _{ij}}{|\mathbf {r} _{ij}|}}$

(2)

$\mathbf {e} _{y}=\mathbf {e} _{z}\times \mathbf {e} _{x}$

(3)

$\mathbf {e} _{kj}=\cos \theta \mathbf {e} _{x}+\sin \theta \mathbf {e} _{y}$

(4)

・微分の導出

○ $\mathbf {r} _{i}$ での微分

${\begin{aligned}{\frac {\partial \cos \theta }{\partial \mathbf {r} _{i}}}&=-\sin \theta {\frac {\partial \theta }{\partial \mathbf {r} _{i}}}\\\end{aligned}}$

構文解析に失敗 (不明な関数「\begin{align}」): {\displaystyle \begin{align} \frac{ \partial \cos \theta }{ \partial x_{i} } &= \frac{ \partial }{ \partial x_{i} } \left( \frac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{ij} | | \mathbf{r}_{kj} | } \right) (\because \theta の定義) \\ &= \frac{ \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{i} } \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{ij} | | \mathbf{r}_{kj} | } + \frac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{kj} | } \frac{ \partial }{ \partial x_{i} } \frac{1}{ | \mathbf{r}_{ij} | }(\because \mathbf{r}_{ij} のみが \mathbf{r}_{i} に依存) \\ &= \frac{ x_{kj} }{ | \mathbf{r}_{ij} | | \mathbf{r}_{kj} | } + \frac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{kj} | } \left( - \frac{1}{ | \mathbf{r}_{ij} |^2 } \right) \frac{ x_{ij} }{ | \mathbf{r}_{ij} | } \\ &= \frac{ | \mathbf{r}_{ij} |^2 x_{kj} - ( \mathbf{r}_{ij} \cdot \mathbf{r}_{kj} ) x_{ij} }{ | \mathbf{r}_{ij} |^3 | \mathbf{r}_{kj} | } = \frac{ ( \mathbf{e}_{kj} )_{x} - ( \mathbf{e}_{ij} \cdot \mathbf{e}_{kj} ) ( \mathbf{e}_{ij} )_{x} }{ | \mathbf{r}_{ij} | } \\ \end{align}}

構文解析に失敗 (不明な関数「\begin{align}」): {\displaystyle \begin{align} \therefore \frac{ \partial \cos \theta }{ \partial \mathbf{r}_{i} } &= \frac{ | \mathbf{r}_{ij} |^2 \mathbf{r}_{kj} - ( \mathbf{r}_{ij} \cdot \mathbf{r}_{kj} ) \mathbf{r}_{ij} }{ | \mathbf{r}_{ij} |^3 | \mathbf{r}_{kj} | } = \frac{ \mathbf{e}_{kj} - ( \mathbf{e}_{ij} \cdot \mathbf{e}_{kj} ) \mathbf{e}_{ij} }{ | \mathbf{r}_{ij} | } \\ &= \frac{ \mathbf{e}_{kj} - \cos \theta \mathbf{e}_{x} }{ | \mathbf{r}_{ij} | } (\because \theta の定義) \\ &= \frac{ \sin \theta \mathbf{e}_{y} }{ | \mathbf{r}_{ij} | } (\because (4)より ) \\ &= \frac{ \sin \theta ( \mathbf{e}_{z} \times \mathbf{e}_{x} ) }{ | \mathbf{r}_{ij} | } (\because (3)より ) \\ \end{align}}

(この変形中で 3, 4 を用いた。)

よって、

${\begin{aligned}{\frac {\partial \theta }{\partial \mathbf {r} _{i}}}&=-{\frac {\mathbf {e} _{kj}-(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})\mathbf {e} _{ij}}{|\mathbf {r} _{ij}|\sin \theta }}\\&=-{\frac {(\mathbf {e} _{z}\times \mathbf {e} _{ij})}{|\mathbf {r} _{ij}|}}\\\end{aligned}}$

○ $\mathbf {r} _{k}$ での微分

i と k を入れ替えれば、$\partial /\partial \mathbf {r} _{i}$ と等価だから、

${\begin{aligned}{\frac {\partial \theta }{\partial \mathbf {r} _{k}}}&=-{\frac {\mathbf {e} _{ij}-(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})\mathbf {e} _{kj}}{|\mathbf {r} _{kj}|\sin \theta }}\\&=-{\frac {(\mathbf {e} _{z}\times \mathbf {e} _{kj})}{|\mathbf {r} _{kj}|}}\\\end{aligned}}$

○ $\mathbf {r} _{j}$ での微分

${\begin{aligned}{\frac {\partial \theta }{\partial \mathbf {r} _{j}}}&=-{\frac {\partial \theta }{\partial \mathbf {r} _{i}}}-{\frac {\partial \theta }{\partial \mathbf {r} _{k}}}\\&={\frac {\mathbf {e} _{kj}-(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})\mathbf {e} _{ij}}{|\mathbf {r} _{ij}|\sin \theta }}+{\frac {\mathbf {e} _{ij}-(\mathbf {e} _{ij}\cdot \mathbf {e} _{kj})\mathbf {e} _{kj}}{|\mathbf {r} _{kj}|\sin \theta }}\\&={\frac {(\mathbf {e} _{z}\times \mathbf {e} _{ij})}{|\mathbf {r} _{ij}|}}+{\frac {(\mathbf {e} _{z}\times \mathbf {e} _{kj})}{|\mathbf {r} _{kj}|}}\\\end{aligned}}$

二面角

・定義

$\phi ={\mbox{sign}}[(\mathbf {r} _{mj}\times \mathbf {r} _{nk})\cdot \mathbf {r} _{kj}]\times \arccos \left({\frac {\mathbf {r} _{mj}\cdot \mathbf {r} _{nk}}{|\mathbf {r} _{mj}||\mathbf {r} _{nk}|}}\right)$

・微分

${\begin{aligned}{\frac {\partial \phi }{\partial \mathbf {r} _{i}}}&={\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|^{2}}}\mathbf {r} _{mj}\\{\frac {\partial \phi }{\partial \mathbf {r} _{l}}}&=-{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{nk}|^{2}}}\mathbf {r} _{nk}\\{\frac {\partial \phi }{\partial \mathbf {r} _{j}}}&=\left({\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right){\frac {\partial \phi }{\partial \mathbf {r} _{i}}}-{\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}{\frac {\partial \phi }{\partial \mathbf {r} _{l}}}\\&=\left({\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right){\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|^{2}}}\mathbf {r} _{mj}-{\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}\left(-{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{nk}|^{2}}}\mathbf {r} _{nk}\right)\\{\frac {\partial \phi }{\partial \mathbf {r} _{k}}}&=\left({\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right){\frac {\partial \phi }{\partial \mathbf {r} _{l}}}-{\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}{\frac {\partial \phi }{\partial \mathbf {r} _{i}}}\\&=\left({\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right)\left(-{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{nk}|^{2}}}\mathbf {r} _{nk}\right)-{\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|^{2}}}\mathbf {r} _{mj}\\\end{aligned}}$

ここで、

$\mathbf {r} _{mj}=\mathbf {r} _{ij}\times \mathbf {r} _{kj}$

(5)

$\mathbf {r} _{nk}=\mathbf {r} _{kj}\times \mathbf {r} _{kl}$

(6)

$\mathbf {e} _{mj}=\cos \phi \mathbf {e} _{nk}+\sin \phi (\mathbf {e} _{nk}\times \mathbf {e} _{kj})$

(7)

$\mathbf {e} _{nk}=\cos \phi \mathbf {e} _{mj}-\sin \phi (\mathbf {e} _{mj}\times \mathbf {e} _{kj})$

(8)

・微分の導出

○ $\mathbf {r} _{i}$ での微分

${\begin{aligned}{\frac {\partial \cos \phi }{\partial \mathbf {r} _{i}}}&=-\sin \phi {\frac {\partial \phi }{\partial \mathbf {r} _{i}}}\\\end{aligned}}$

構文解析に失敗 (不明な関数「\begin{align}」): {\displaystyle \begin{align} \frac{ \partial \cos \phi }{ \partial x_{i} } &= \frac{ \partial }{ \partial x_{i} } \left( \frac{ \mathbf{r}_{mj} \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{mj} | | \mathbf{r}_{nk} | } \right) (\because \phi の定義 ) \\ &= \frac{ \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{i} } \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{mj} | | \mathbf{r}_{nk} | } + \frac{ \mathbf{r}_{mj} \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{nk} | } \frac{ \partial }{ \partial x_{i} } \frac{1}{ | \mathbf{r}_{mj} | } (\because \mathbf{r}_{mj} のみが \mathbf{r}_{i} に依存) \\ &= \frac{ \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{i} } \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{mj} | | \mathbf{r}_{nk} | } + \frac{ \mathbf{r}_{mj} \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{nk} | } \left( - \frac{1}{ | \mathbf{r}_{mj} |^2 } \right) \frac{ \partial | \mathbf{r}_{mj} | }{ \partial x_{i} } \\ &= \frac{ \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{i} } \cdot \mathbf{e}_{nk} - \cos \phi \frac{ \partial | \mathbf{r}_{mj} | }{ \partial x_{i} } }{ | \mathbf{r}_{mj} | } \\ &= \frac{ \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{i} } \cdot \mathbf{e}_{nk} - \cos \phi \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{i} } \cdot \mathbf{e}_{mj} }{ | \mathbf{r}_{mj} | } (\because (9)より) \\ &= \frac{ - \sin \phi \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{i} } \cdot ( \mathbf{e}_{mj} \times \mathbf{e}_{kj} ) }{ | \mathbf{r}_{mj} | } (\because (8)より) \\ &= - \sin \phi \frac{ \left( \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{i} } \times \mathbf{r}_{kj} \right) \cdot ( \mathbf{e}_{mj} \times \mathbf{e}_{kj} ) }{ | \mathbf{r}_{mj} | } (\because (5)より) \\ &= - \sin \phi \frac{ \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{i} } \cdot [ \mathbf{r}_{kj} \times ( \mathbf{e}_{mj} \times \mathbf{e}_{kj} ) ] }{ | \mathbf{r}_{mj} | } (\because スカラー三重積(10)より) \\ &= - \sin \phi \frac{ \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{i} } \cdot [ ( \mathbf{r}_{kj} \cdot \mathbf{e}_{kj} ) \mathbf{e}_{mj} - ( \mathbf{r}_{kj} \cdot \mathbf{e}_{mj} ) \mathbf{e}_{kj} ] }{ | \mathbf{r}_{mj} | } (\because ベクトル三重積(11)より) \\ &= - \sin \phi \frac{ | \mathbf{r}_{kj} | \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{i} } \cdot \mathbf{e}_{mj} }{ | \mathbf{r}_{mj} | } (\because \mathbf{r}_{kj} \perp \mathbf{e}_{mj}) \\ &= - \sin \phi \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{mj} | } ( \mathbf{e}_{mj} )_{x} \\ \end{align}}

(この変形中で 5, 8, 9, 10, 11 を用いた。

${\frac {\partial |\mathbf {r} _{mj}|}{\partial x_{i}}}={\frac {\partial \mathbf {r} _{mj}}{\partial x_{i}}}\cdot \mathbf {e} _{mj}$

(9)

${\begin{aligned}\because {\frac {\partial |\mathbf {r} _{mj}|}{\partial x_{i}}}&={\frac {\partial {\sqrt {(\mathbf {r} _{mj})_{x}^{2}+(\mathbf {r} _{mj})_{y}^{2}+(\mathbf {r} _{mj})_{z}^{2}}}}{\partial x_{i}}}\\&={\frac {2\left((\mathbf {r} _{mj})_{x}\cdot {\frac {\partial (\mathbf {r} _{mj})_{x}}{\partial x_{i}}}+(\mathbf {r} _{mj})_{y}\cdot {\frac {\partial (\mathbf {r} _{mj})_{y}}{\partial x_{i}}}+(\mathbf {r} _{mj})_{z}\cdot {\frac {\partial (\mathbf {r} _{mj})_{z}}{\partial x_{i}}}\right)}{2{\sqrt {(\mathbf {r} _{mj})_{x}^{2}+(\mathbf {r} _{mj})_{y}^{2}+(\mathbf {r} _{mj})_{z}^{2}}}}}\\&={\frac {{\frac {\partial \mathbf {r} _{mj}}{\partial x_{i}}}\cdot \mathbf {r} _{mj}}{|\mathbf {r} _{mj}|}}\\&={\frac {\partial \mathbf {r} _{mj}}{\partial x_{i}}}\cdot \mathbf {e} _{mj}\\\end{aligned}}$

$\mathbf {A} \cdot (\mathbf {B} \times \mathbf {C} )=\mathbf {B} \cdot (\mathbf {C} \times \mathbf {A} )=\mathbf {C} \cdot (\mathbf {A} \times \mathbf {B} )=-\mathbf {A} \cdot (\mathbf {C} \times \mathbf {B} )=-\mathbf {A} \cdot (\mathbf {C} \times \mathbf {B} )=-\mathbf {B} \cdot (\mathbf {A} \times \mathbf {C} )=-\mathbf {C} \cdot (\mathbf {B} \times \mathbf {A} )$

(10)

$\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {A} \cdot \mathbf {B} )\mathbf {C}$

(11)

)

${\begin{aligned}\therefore {\frac {\partial \cos \phi }{\partial \mathbf {r} _{i}}}&=-\sin \phi {\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|}}\mathbf {e} _{mj}\\\end{aligned}}$

よって、

${\begin{aligned}{\frac {\partial \phi }{\partial \mathbf {r} _{i}}}&={\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|}}\mathbf {e} _{mj}\\&={\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|^{2}}}\mathbf {r} _{mj}\\\end{aligned}}$

○ $\mathbf {r} _{l}$ での微分

i と l , k と j , m と n を入れ替えれば、$\partial /\partial \mathbf {r} _{i}$ と等価だから、$\mathbf {r} _{mj}$ → $\mathbf {r} _{nk}$ の変換の際の符号に気をつけて

${\begin{aligned}{\frac {\partial \phi }{\partial \mathbf {r} _{l}}}&=-{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{nk}|}}\mathbf {e} _{nk}\\&=-{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{nk}|^{2}}}\mathbf {r} _{nk}\\\end{aligned}}$

○ $\mathbf {r} _{j}$ での微分

${\begin{aligned}{\frac {\partial \cos \phi }{\partial \mathbf {r} _{j}}}&=-\sin \phi {\frac {\partial \phi }{\partial \mathbf {r} _{j}}}\\\end{aligned}}$

構文解析に失敗 (不明な関数「\begin{align}」): {\displaystyle \begin{align} \frac{ \partial \cos \phi }{ \partial x_{j} } &= \frac{ \partial }{ \partial x_{j} } \left( \frac{ \mathbf{r}_{mj} \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{mj} | | \mathbf{r}_{nk} | } \right) (\because \phi の定義) \\ &= \frac{ \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{j} } \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{mj} | | \mathbf{r}_{nk} | } + \frac{ \mathbf{r}_{mj} \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{nk} | } \frac{ \partial }{ \partial x_{j} } \frac{1}{ | \mathbf{r}_{mj} | } + \frac{ \frac{ \partial \mathbf{r}_{nk} }{ \partial x_{j} } \cdot \mathbf{r}_{mj} }{ | \mathbf{r}_{mj} | | \mathbf{r}_{nk} | } + \frac{ \mathbf{r}_{mj} \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{mj} | } \frac{ \partial }{ \partial x_{j} } \frac{1}{ | \mathbf{r}_{nk} | } (\because \mathbf{r}_{mj}, \mathbf{r}_{nk} 共に \mathbf{r}_{i} に依存) \\ &= \frac{ \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{j} } \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{mj} | | \mathbf{r}_{nk} | } + \frac{ \mathbf{r}_{mj} \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{nk} | } \left( - \frac{1}{ | \mathbf{r}_{mj} |^2 } \right) \frac{ \partial | \mathbf{r}_{mj} | }{ \partial x_{j} } + \frac{ \frac{ \partial \mathbf{r}_{nk} }{ \partial x_{j} } \cdot \mathbf{r}_{mj} }{ | \mathbf{r}_{mj} | | \mathbf{r}_{nk} | } + \frac{ \mathbf{r}_{mj} \cdot \mathbf{r}_{nk} }{ | \mathbf{r}_{mj} | } \left( - \frac{1}{ | \mathbf{r}_{nk} |^2 } \right) \frac{ \partial | \mathbf{r}_{nk} | }{ \partial x_{j} } \\ &= \frac{ \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{j} } \cdot \mathbf{e}_{nk} - \cos \phi \frac{ \partial | \mathbf{r}_{mj} | }{ \partial x_{j} } }{ | \mathbf{r}_{mj} | } + \frac{ \frac{ \partial \mathbf{r}_{nk} }{ \partial x_{j} } \cdot \mathbf{e}_{mj} - \cos \phi \frac{ \partial | \mathbf{r}_{nk} | }{ \partial x_{j} } }{ | \mathbf{r}_{nk} | } \\ &= \frac{ \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{j} } \cdot \mathbf{e}_{nk} - \cos \phi \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{j} } \cdot \mathbf{e}_{mj} }{ | \mathbf{r}_{mj} | } + \frac{ \frac{ \partial \mathbf{r}_{nk} }{ \partial x_{j} } \cdot \mathbf{e}_{mj} - \cos \phi \frac{ \partial \mathbf{r}_{nk} }{ \partial x_{j} } \cdot \mathbf{e}_{nk} }{ | \mathbf{r}_{nk} | } (\because (9), (12)より) \\ &= \frac{ - \sin \phi \frac{ \partial \mathbf{r}_{mj} }{ \partial x_{j} } \cdot ( \mathbf{e}_{mj} \times \mathbf{e}_{kj} ) }{ | \mathbf{r}_{mj} | } + \frac{ \sin \phi \frac{ \partial \mathbf{r}_{nk} }{ \partial x_{j} } \cdot ( \mathbf{e}_{nk} \times \mathbf{e}_{kj} ) }{ | \mathbf{r}_{nk} | } \\ &= - \sin \phi \frac{ \left( \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{j} } \times \mathbf{r}_{kj} + \mathbf{r}_{ij} \times \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{j} } \right) \cdot ( \mathbf{e}_{mj} \times \mathbf{e}_{kj} ) }{ | \mathbf{r}_{mj} | } + \sin \phi \frac{ \left( \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{j} } \times \mathbf{r}_{kl} \right) \cdot ( \mathbf{e}_{nk} \times \mathbf{e}_{kj} ) }{ | \mathbf{r}_{nk} | } (\because (7), (8)より) \\ &= - \sin \phi \frac{ \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{j} } \cdot [ \mathbf{r}_{kj} \times ( \mathbf{e}_{mj} \times \mathbf{e}_{kj} ) ] - \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{j} } \cdot [ \mathbf{r}_{ij} \times ( \mathbf{e}_{mj} \times \mathbf{e}_{kj} ) ] }{ | \mathbf{r}_{mj} | } + \sin \phi \frac{ \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{j} } \cdot [ \mathbf{r}_{kl} \times ( \mathbf{e}_{nk} \times \mathbf{e}_{kj} ) ] }{ | \mathbf{r}_{nk} | } (\because スカラー三重積(10)より) \\ &= - \sin \phi \frac{ \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{j} } \cdot [ ( \mathbf{r}_{kj} \cdot \mathbf{e}_{kj} ) \mathbf{e}_{mj} - ( \mathbf{r}_{kj} \cdot \mathbf{e}_{mj} ) \mathbf{e}_{kj} ] - \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{j} } \cdot [ ( \mathbf{r}_{ij} \cdot \mathbf{e}_{kj} ) \mathbf{e}_{mj} - ( \mathbf{r}_{ij} \cdot \mathbf{e}_{mj} ) \mathbf{e}_{kj} ]}{ | \mathbf{r}_{mj} | } + \sin \phi \frac{ \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{j} } \cdot [ ( \mathbf{r}_{kl} \cdot \mathbf{e}_{kj} ) \mathbf{e}_{nk} - ( \mathbf{r}_{kl} \cdot \mathbf{e}_{nk} ) \mathbf{e}_{kj} ] }{ | \mathbf{r}_{nk} | } (\because ベクトル三重積(11)より) \\ &= - \sin \phi \left( \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{mj} |^2 } \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{j} } \cdot \mathbf{r}_{mj} - \frac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{kj} |^2 } \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{mj} |^2 } \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{j} } \cdot \mathbf{r}_{mj} \right) + \sin \phi \frac{ \mathbf{r}_{kl} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{kj} |^2 } \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{nk} |^2 } \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{j} } \cdot \mathbf{r}_{nk} (\because \mathbf{r}_{kj} \perp \mathbf{e}_{mj}, \because \mathbf{r}_{ij} \perp \mathbf{e}_{mj}) \\ &= - \sin \phi \left( \frac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{kj} |^2 } \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{mj} |^2 } ( \mathbf{r}_{mj} )_{x} - \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{mj} |^2 } ( \mathbf{r}_{mj} )_{x} \right) - \sin \phi \frac{ \mathbf{r}_{kl} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{kj} |^2 } \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{nk} |^2 } ( \mathbf{r}_{nk} )_{x} \\ &= - \sin \phi \left( \frac{ \mathbf{r}_{ij} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{kj} |^2 } - 1 \right) \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{mj} |^2 } ( \mathbf{r}_{mj} )_{x} - \sin \phi \frac{ \mathbf{r}_{kl} \cdot \mathbf{r}_{kj} }{ | \mathbf{r}_{kj} |^2 } \frac{ | \mathbf{r}_{kj} | }{ | \mathbf{r}_{nk} |^2 } ( \mathbf{r}_{nk} )_{x} \\ \end{align}}

(この変形中で 7, 8, 9, 10, 11, 12 を用いた。

${\frac {\partial |\mathbf {r} _{nk}|}{\partial x_{i}}}={\frac {\partial \mathbf {r} _{nk}}{\partial x_{i}}}\cdot \mathbf {e} _{nk}$

(12)

)

${\begin{aligned}\therefore {\frac {\partial \cos \phi }{\partial \mathbf {r} _{j}}}&=-\sin \phi \left({\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right){\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|^{2}}}\mathbf {r} _{mj}-\sin \phi {\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{nk}|^{2}}}\mathbf {r} _{nk}\\\end{aligned}}$

よって、

${\begin{aligned}{\frac {\partial \phi }{\partial \mathbf {r} _{j}}}&=\left({\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right){\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|^{2}}}\mathbf {r} _{mj}+{\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{nk}|^{2}}}\mathbf {r} _{nk}\\&=\left({\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right){\frac {\partial \phi }{\partial \mathbf {r} _{i}}}-{\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}{\frac {\partial \phi }{\partial \mathbf {r} _{l}}}\\\end{aligned}}$

○ $\mathbf {r} _{k}$ での微分

i と l , k と j , m と n を入れ替えれば、$\partial /\partial \mathbf {r} _{j}$ と等価だから

${\begin{aligned}{\frac {\partial \phi }{\partial \mathbf {r} _{k}}}&=-\left({\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right){\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{nk}|^{2}}}\mathbf {r} _{nk}-{\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}{\frac {|\mathbf {r} _{kj}|}{|\mathbf {r} _{mj}|^{2}}}\mathbf {r} _{mj}\\&=\left({\frac {\mathbf {r} _{kl}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}-1\right){\frac {\partial \phi }{\partial \mathbf {r} _{l}}}-{\frac {\mathbf {r} _{ij}\cdot \mathbf {r} _{kj}}{|\mathbf {r} _{kj}|^{2}}}{\frac {\partial \phi }{\partial \mathbf {r} _{i}}}\\\end{aligned}}$

面外角(Wagging angle)

・定義

$\chi =\arcsin(\mathbf {e} _{ij}\cdot \mathbf {e} _{z})=\arcsin \left({\frac {\mathbf {r} _{ij}\cdot (\mathbf {r} _{kj}\times \mathbf {r} _{lj})}{|\mathbf {r} _{ij}||\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\right)$

・微分

${\begin{aligned}{\frac {\partial \chi }{\partial \mathbf {r} _{i}}}&=-{\frac {(\mathbf {e} _{y}\times \mathbf {e} _{ij})}{|\mathbf {r} _{ij}|}}\\&=-{\frac {(\mathbf {e} _{y}\times \mathbf {r} _{ij})}{|\mathbf {r} _{ij}|^{2}}}\\{\frac {\partial \chi }{\partial \mathbf {r} _{j}}}&=-{\frac {\partial \chi }{\partial \mathbf {r} _{i}}}-{\frac {\partial \chi }{\partial \mathbf {r} _{k}}}-{\frac {\partial \chi }{\partial \mathbf {r} _{l}}}\\&={\frac {(\mathbf {e} _{y}\times \mathbf {e} _{ij})}{|\mathbf {r} _{ij}|}}-{\frac {(\mathbf {e} _{x}\times \mathbf {r} _{kl})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\&={\frac {(\mathbf {e} _{y}\times \mathbf {r} _{ij})}{|\mathbf {r} _{ij}|^{2}}}-{\frac {(\mathbf {e} _{x}\times \mathbf {r} _{kl})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\{\frac {\partial \chi }{\partial \mathbf {r} _{k}}}&=-{\frac {(\mathbf {e} _{x}\times \mathbf {r} _{lj})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\{\frac {\partial \chi }{\partial \mathbf {r} _{l}}}&={\frac {(\mathbf {e} _{x}\times \mathbf {r} _{kj})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\\end{aligned}}$

ここで、

$\mathbf {e} _{z}={\frac {\mathbf {r} _{kj}\times \mathbf {r} _{lj}}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}$

(13)

$\mathbf {e} _{y}={\frac {\mathbf {e} _{z}\times \mathbf {e} _{ij}}{|\mathbf {e} _{z}\times \mathbf {e} _{ij}|}}$

(14)

$\mathbf {e} _{x}=\mathbf {e} _{y}\times \mathbf {e} _{z}$

(15)

$\mathbf {e} _{ij}=\cos \left({\frac {\pi }{2}}-\chi \right)\mathbf {e} _{z}+\sin \left({\frac {\pi }{2}}-\chi \right)\mathbf {e} _{x}=\sin \chi \mathbf {e} _{z}+\cos \chi \mathbf {e} _{x}$

(16)

・微分の導出

○ $\mathbf {r} _{i}$ での微分

${\begin{aligned}{\frac {\partial \sin \chi }{\partial \mathbf {r} _{i}}}&=\cos \chi {\frac {\partial \chi }{\partial \mathbf {r} _{i}}}\\\end{aligned}}$

構文解析に失敗 (不明な関数「\begin{align}」): {\displaystyle \begin{align} \frac{ \partial \sin \chi }{ \partial x_{i} } &= \frac{ \partial }{ \partial x_{i} } \left( \frac{ \mathbf{r}_{ij} \cdot ( \mathbf{r}_{kj} \times \mathbf{r}_{lj} ) }{ | \mathbf{r}_{ij} | | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \right) (\because \chi の定義) \\ &= \frac{ \frac{ \partial \mathbf{r}_{ij} }{ \partial x_{i} } \cdot ( \mathbf{r}_{kj} \times \mathbf{r}_{lj} ) }{ | \mathbf{r}_{ij} | | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } + \frac{ \mathbf{r}_{ij} \cdot ( \mathbf{r}_{kj} \times \mathbf{r}_{lj} ) }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \frac{ \partial }{ \partial x_{i} } \frac{1}{ | \mathbf{r}_{ij} | } (\because \mathbf{r}_{ij} のみが \mathbf{r}_{i} に依存) \\ &= \frac{ ( \mathbf{r}_{kj} \times \mathbf{r}_{lj} )_{x} }{ | \mathbf{r}_{ij} | | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } + \frac{ \mathbf{r}_{ij} \cdot ( \mathbf{r}_{kj} \times \mathbf{r}_{lj} ) }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \left( - \frac{1}{ | \mathbf{r}_{ij} |^2 } \right) \frac{ x_{ij} }{ | \mathbf{r}_{ij} | } \\ &= \frac{ ( \mathbf{e}_{z} )_{x} - ( \mathbf{e}_{ij} \cdot \mathbf{e}_{z} ) ( \mathbf{e}_{ij} )_{x} }{ | \mathbf{r}_{ij} | } \\ \end{align}}

構文解析に失敗 (不明な関数「\begin{align}」): {\displaystyle \begin{align} \therefore \frac{ \partial \sin \chi }{ \partial \mathbf{r}_{i} } &= \frac{ \mathbf{e}_{z} - ( \mathbf{e}_{ij} \cdot \mathbf{e}_{z} ) \mathbf{e}_{ij} }{ | \mathbf{r}_{ij} | } \\ &= \frac{ ( \mathbf{e}_{ij} \times \mathbf{e}_{z} ) \times \mathbf{e}_{ij} }{ | \mathbf{r}_{ij} | } (\because ベクトル三重積(17)より) \\ &= \frac{ ( [ \sin \chi \mathbf{e}_{z} + \cos \chi \mathbf{e}_{x} ] \times \mathbf{e}_{z} ) \times \mathbf{e}_{ij} }{ | \mathbf{r}_{ij} | } (\because (16)より) \\ &= \frac{ - \cos \chi ( \mathbf{e}_{y} \times \mathbf{e}_{ij} ) }{ | \mathbf{r}_{ij} | } \\ \end{align}}

(この変形中で 16, 17 を用いた。

$(\mathbf {A} \times \mathbf {B} )\times \mathbf {C} =(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {B} \cdot \mathbf {C} )\mathbf {A}$

(17)

)

よって、

${\begin{aligned}{\frac {\partial \chi }{\partial \mathbf {r} _{i}}}&=-{\frac {(\mathbf {e} _{y}\times \mathbf {e} _{ij})}{|\mathbf {r} _{ij}|}}\\&=-{\frac {(\mathbf {e} _{y}\times \mathbf {r} _{ij})}{|\mathbf {r} _{ij}|^{2}}}\\\end{aligned}}$

○ $\mathbf {r} _{k}$ での微分

${\begin{aligned}{\frac {\partial \sin \chi }{\partial \mathbf {r} _{k}}}&=\cos \chi {\frac {\partial \chi }{\partial \mathbf {r} _{k}}}\\\end{aligned}}$

構文解析に失敗 (不明な関数「\begin{align}」): {\displaystyle \begin{align} \frac{ \partial \sin \chi }{ \partial x_{k} } &= \frac{ \partial }{ \partial x_{k} } \left( \frac{ \mathbf{r}_{ij} \cdot ( \mathbf{r}_{kj} \times \mathbf{r}_{lj} ) }{ | \mathbf{r}_{ij} | | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \right) (\because \chi の定義) \\ &= \frac{ \mathbf{r}_{ij} \cdot \left( \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{k} } \times \mathbf{r}_{lj} \right) }{ | \mathbf{r}_{ij} | | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } + \frac{ \mathbf{r}_{ij} \cdot ( \mathbf{r}_{kj} \times \mathbf{r}_{lj} ) }{ | \mathbf{r}_{ij} | } \frac{ \partial }{ \partial x_{k} } \frac{1}{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } (\because \mathbf{r}_{kj} のみが \mathbf{r}_{k} に依存) \\ &= \frac{ \mathbf{e}_{ij} \cdot \left( \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{k} } \times \mathbf{r}_{lj} \right) }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } + \mathbf{e}_{ij} \cdot ( | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | \mathbf{e}_{z} ) \left( - \frac{1}{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} |^2 } \right) \frac{ \partial | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | }{ \partial x_{k} } (\because (13)より) \\ &= \frac{ \frac{ \partial \mathbf{r}_{kj} }{ \partial x_{k} } \cdot ( \mathbf{r}_{lj} \times \mathbf{e}_{ij} ) }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } + ( \mathbf{e}_{ij} \cdot \mathbf{e}_{z} ) | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | \left( - \frac{1}{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} |^2 } \right) \frac{ | \mathbf{r}_{lj} |^2 x_{kj} - ( \mathbf{r}_{kj} \cdot \mathbf{r}_{lj} ) x_{lj}}{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } (\because (10),(18) より) \\ &= \frac{ ( \mathbf{r}_{lj} \times \mathbf{e}_{ij} )_{x} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } - \sin \chi \left( \frac{ | \mathbf{r}_{lj} |^2 x_{kj} - ( \mathbf{r}_{kj} \cdot \mathbf{r}_{lj} ) x_{lj} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} |^2 } \right) (\because \chi の定義) \\ \end{align}}

(この変形中で 10, 13, 18 を用いた。

${\frac {\partial |\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}{\partial x_{k}}}={\frac {|\mathbf {r} _{lj}|^{2}x_{kj}-(\mathbf {r} _{kj}\cdot \mathbf {r} _{lj})x_{lj}}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}$

(18)

${\begin{aligned}\because {\frac {\partial |\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}{\partial x_{k}}}&={\frac {\partial }{\partial x_{k}}}{\sqrt {(y_{kj}z_{lj}-z_{kj}y_{lj})^{2}+(z_{kj}x_{lj}-x_{kj}z_{lj})^{2}+(x_{kj}y_{lj}-y_{kj}x_{lj})^{2}}}\\&={\frac {-2z_{lj}(z_{kj}x_{lj}-x_{kj}z_{lj})+2y_{lj}(x_{kj}y_{lj}-y_{kj}x_{lj})}{2{\sqrt {(y_{kj}z_{lj}-z_{kj}y_{lj})^{2}+(z_{kj}x_{lj}-x_{kj}z_{lj})^{2}+(x_{kj}y_{lj}-y_{kj}x_{lj})^{2}}}}}\\&={\frac {x_{kj}(y_{lj}^{2}+z_{lj}^{2})-x_{lj}(y_{kj}y_{lj}+z_{kj}z_{lj})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\&={\frac {x_{kj}(x_{lj}^{2}+y_{lj}^{2}+z_{lj}^{2})-x_{lj}(x_{kj}x_{lj}+y_{kj}y_{lj}+z_{kj}z_{lj})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\&={\frac {|\mathbf {r} _{lj}|^{2}x_{kj}-(\mathbf {r} _{kj}\cdot \mathbf {r} _{lj})x_{lj}}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\\end{aligned}}$

)

構文解析に失敗 (不明な関数「\begin{align}」): {\displaystyle \begin{align} \therefore \frac{ \partial \sin \chi }{ \partial \mathbf{r}_{k} } &= \frac{ \mathbf{r}_{lj} \times \mathbf{e}_{ij} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } - \sin \chi \left( \frac{ | \mathbf{r}_{lj} |^2 \mathbf{r}_{kj} - ( \mathbf{r}_{kj} \cdot \mathbf{r}_{lj} ) \mathbf{r}_{lj} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} |^2 } \right) \\ &= \frac{ \mathbf{r}_{lj} \times \mathbf{e}_{ij} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } - \sin \chi \left( \frac{ \mathbf{r}_{lj} \times ( \mathbf{r}_{kj} \times \mathbf{r}_{lj} ) }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} |^2 } \right) (\because ベクトル三重積(11)より) \\ &= \frac{ \mathbf{r}_{lj} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \times \left( \mathbf{e}_{ij} - \sin \chi \frac{ \mathbf{r}_{kj} \times \mathbf{r}_{lj} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \right) (\because (13)より) \\ &= \frac{ \mathbf{r}_{lj} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \times ( \mathbf{e}_{ij} - \sin \chi \mathbf{e}_{z} ) \\ &= \frac{ \mathbf{r}_{lj} }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \times ( \cos \chi \mathbf{e}_{x} ) \\ &= - \cos \chi \frac{ ( \mathbf{e}_{x} \times \mathbf{r}_{lj} ) }{ | \mathbf{r}_{kj} \times \mathbf{r}_{lj} | } \\ \end{align}}

よって、

${\begin{aligned}{\frac {\partial \chi }{\partial \mathbf {r} _{k}}}&=-{\frac {(\mathbf {e} _{x}\times \mathbf {r} _{lj})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\\end{aligned}}$

○ $\mathbf {r} _{l}$ での微分

k と l を入れ替えれば、$\partial /\partial \mathbf {r} _{k}$ と等価だから、符号に気をつけて

${\begin{aligned}{\frac {\partial \chi }{\partial \mathbf {r} _{l}}}&={\frac {(\mathbf {e} _{x}\times \mathbf {r} _{kj})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\\end{aligned}}$

○ $\mathbf {r} _{j}$ での微分

${\begin{aligned}{\frac {\partial \chi }{\partial \mathbf {r} _{j}}}&=-{\frac {\partial \chi }{\partial \mathbf {r} _{i}}}-{\frac {\partial \chi }{\partial \mathbf {r} _{k}}}-{\frac {\partial \chi }{\partial \mathbf {r} _{l}}}\\&={\frac {(\mathbf {e} _{y}\times \mathbf {r} _{ij})}{|\mathbf {r} _{ij}|^{2}}}+{\frac {(\mathbf {e} _{x}\times \mathbf {r} _{lj})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}-{\frac {(\mathbf {e} _{x}\times \mathbf {r} _{kj})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\&={\frac {(\mathbf {e} _{y}\times \mathbf {r} _{ij})}{|\mathbf {r} _{ij}|^{2}}}-{\frac {(\mathbf {e} _{x}\times \mathbf {r} _{kl})}{|\mathbf {r} _{kj}\times \mathbf {r} _{lj}|}}\\\end{aligned}}$