スケール因子・微小線素と単位基底ベクトル・ベクトルの成分同士の関係
スケール因子・微小線素と単位基底ベクトル・ベクトルの成分同士の関係
スケール因子・微小線素と単位基底ベクトル・ベクトルの成分同士について次が成り立つ。
\[ \boldsymbol{u}_{i}=\frac{1}{h_{i}}\frac{\partial\boldsymbol{r}}{\partial q_{i}} \] \[ \boldsymbol{e}_{k}=\sum_{i}\frac{\partial q_{i}}{\partial x_{k}}h_{i}\boldsymbol{u}_{i} \] ここで\(h_{i}\)はスケール因子
\begin{align*} h_{i} & =\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{i}}\right\Vert \\ & =\sqrt{\sum_{j}\frac{\partial x_{j}}{\partial q_{i}}\frac{\partial x_{j}}{\partial q_{i}}} \end{align*} である。
\[ h_{i}h_{j}\delta_{ij}=\sum_{k}\frac{\partial x_{k}}{\partial q_{i}}\frac{\partial x_{k}}{\partial q_{j}} \]
\[ ds^{2}=\sum_{j}h_{j}^{2}dq_{j}^{2} \]
\[ A_{k}'=\sum_{j}A_{j}\frac{\partial q_{k}}{\partial x_{j}}h_{k} \] \[ A_{k}=\sum_{j}A_{j}'\frac{1}{h_{j}}\frac{\partial x_{k}}{\partial q_{j}} \] となる。
すなわち、ヤコビ行列を\(J\)とすると、
\[ \left|\det J\right|=\prod_{k=1}^{n}h_{k} \] となる。
スケール因子・微小線素と単位基底ベクトル・ベクトルの成分同士について次が成り立つ。
(1)
直交曲線座標\(x_{i}\)の基底ベクトルを\(\boldsymbol{e}_{i}\)として直交曲線座標\(q_{i}\)の単位基底ベクトル\(\boldsymbol{u}_{i}\)とすると次の関係がある。\[ \boldsymbol{u}_{i}=\frac{1}{h_{i}}\frac{\partial\boldsymbol{r}}{\partial q_{i}} \] \[ \boldsymbol{e}_{k}=\sum_{i}\frac{\partial q_{i}}{\partial x_{k}}h_{i}\boldsymbol{u}_{i} \] ここで\(h_{i}\)はスケール因子
\begin{align*} h_{i} & =\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{i}}\right\Vert \\ & =\sqrt{\sum_{j}\frac{\partial x_{j}}{\partial q_{i}}\frac{\partial x_{j}}{\partial q_{i}}} \end{align*} である。
(2)
スケール因子について以下が成り立つ。\[ h_{i}h_{j}\delta_{ij}=\sum_{k}\frac{\partial x_{k}}{\partial q_{i}}\frac{\partial x_{k}}{\partial q_{j}} \]
(3)
直交曲線座標\(q_{i}\)の微小線素\(ds\)は以下のようになる。\[ ds^{2}=\sum_{j}h_{j}^{2}dq_{j}^{2} \]
(4)
直交座標でのベクトルの成分を\(A_{k}\)、直交曲線座標でのベクトルの成分を\(A_{k}'\)とすると、\[ A_{k}'=\sum_{j}A_{j}\frac{\partial q_{k}}{\partial x_{j}}h_{k} \] \[ A_{k}=\sum_{j}A_{j}'\frac{1}{h_{j}}\frac{\partial x_{k}}{\partial q_{j}} \] となる。
(5)
スケール因子の全ての積と直交座標系から直交曲線座標系へのヤコビアンの絶対値は等しい。すなわち、ヤコビ行列を\(J\)とすると、
\[ \left|\det J\right|=\prod_{k=1}^{n}h_{k} \] となる。
極座標
極座標\[ \begin{cases} x=r\sin\theta\cos\phi\\ y=r\sin\theta\sin\phi\\ z=r\cos\theta \end{cases} \] \[ \begin{cases} r=\sqrt{x^{2}+y^{2}+z^{2}}\\ \theta=\cos^{\bullet}\left(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\right)\\ \phi=\tan^{\bullet}\left(\frac{y}{x}\right) \end{cases} \] のとき、
\[ \left(\begin{array}{ccc} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial\theta} & \frac{\partial x}{\partial\phi}\\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial\theta} & \frac{\partial y}{\partial\phi}\\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial\theta} & \frac{\partial z}{\partial\phi} \end{array}\right)=\left(\begin{array}{ccc} \sin\theta\cos\phi & r\cos\theta\cos\phi & -r\sin\theta\sin\phi\\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi\\ \cos\theta & -r\sin\theta & 0 \end{array}\right) \] \[ \left(\begin{array}{ccc} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} & \frac{\partial r}{\partial z}\\ \frac{\partial\theta}{\partial x} & \frac{\partial\theta}{\partial y} & \frac{\partial\theta}{\partial z}\\ \frac{\partial\phi}{\partial x} & \frac{\partial\phi}{\partial y} & \frac{\partial\phi}{\partial z} \end{array}\right)=\left(\begin{array}{ccc} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta\\ \frac{\cos\theta\cos\phi}{r} & \frac{\cos\theta\sin\phi}{r} & -\frac{\sin\theta}{r}\\ -\frac{\sin\phi}{r\sin\theta} & \frac{\cos\phi}{r\sin\theta} & 0 \end{array}\right) \] \[ \begin{cases} h_{r}=1\\ h_{\theta}=r\\ h_{\phi}=r\sin\theta \end{cases} \] となる。
これより、基底ベクトル同士の関係は
\begin{align*} \left(\begin{array}{c} \boldsymbol{e}_{r}\\ \boldsymbol{e}_{\theta}\\ \boldsymbol{e}_{\phi} \end{array}\right) & =\left(\begin{array}{ccc} \frac{1}{h_{1}}\frac{\partial x}{\partial r} & \frac{1}{h_{1}}\frac{\partial y}{\partial r} & \frac{1}{h_{1}}\frac{\partial z}{\partial r}\\ \frac{1}{h_{2}}\frac{\partial x}{\partial\theta} & \frac{1}{h_{2}}\frac{\partial y}{\partial\theta} & \frac{1}{h_{2}}\frac{\partial z}{\partial\theta}\\ \frac{1}{h_{3}}\frac{\partial x}{\partial\phi} & \frac{1}{h_{3}}\frac{\partial y}{\partial\phi} & \frac{1}{h_{3}}\frac{\partial z}{\partial\phi} \end{array}\right)\left(\begin{array}{c} \boldsymbol{i}\\ \boldsymbol{j}\\ \boldsymbol{k} \end{array}\right)\\ & =\left(\begin{array}{ccc} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta\\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta\\ -\sin\phi & \cos\phi & 0 \end{array}\right)\left(\begin{array}{c} \boldsymbol{i}\\ \boldsymbol{j}\\ \boldsymbol{k} \end{array}\right) \end{align*} \begin{align*} \left(\begin{array}{c} \boldsymbol{i}\\ \boldsymbol{j}\\ \boldsymbol{k} \end{array}\right) & =\left(\begin{array}{ccc} h_{1}\frac{\partial r}{\partial x} & h_{2}\frac{\partial\theta}{\partial x} & h_{3}\frac{\partial\phi}{\partial x}\\ h_{1}\frac{\partial r}{\partial y} & h_{2}\frac{\partial\theta}{\partial y} & h_{3}\frac{\partial\phi}{\partial y}\\ h_{1}\frac{\partial r}{\partial z} & h_{2}\frac{\partial\theta}{\partial z} & h_{3}\frac{\partial\phi}{\partial z} \end{array}\right)\left(\begin{array}{c} \boldsymbol{e}_{r}\\ \boldsymbol{e}_{\theta}\\ \boldsymbol{e}_{\phi} \end{array}\right)\\ & =\left(\begin{array}{ccc} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi\\ \cos\phi & -\sin\theta & 0 \end{array}\right)\left(\begin{array}{c} \boldsymbol{e}_{r}\\ \boldsymbol{e}_{\theta}\\ \boldsymbol{e}_{\phi} \end{array}\right) \end{align*} となる。
また逆行列を使って
\begin{align*} \left(\begin{array}{c} \boldsymbol{i}\\ \boldsymbol{j}\\ \boldsymbol{k} \end{array}\right) & =\left(\begin{array}{ccc} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta\\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta\\ -\sin\phi & \cos\phi & 0 \end{array}\right)^{-1}\left(\begin{array}{c} \boldsymbol{e}_{r}\\ \boldsymbol{e}_{\theta}\\ \boldsymbol{e}_{\phi} \end{array}\right)\\ & =\left(\begin{array}{ccc} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi\\ \cos\phi & -\sin\theta & 0 \end{array}\right)\left(\begin{array}{c} \boldsymbol{e}_{r}\\ \boldsymbol{e}_{\theta}\\ \boldsymbol{e}_{\phi} \end{array}\right) \end{align*} としても求まる。
ベクトルの成分同士の関係は
\begin{align*} \left(\begin{array}{c} A_{r}\\ A_{\theta}\\ A_{\phi} \end{array}\right) & =\left(\begin{array}{ccc} h_{1}\frac{\partial r}{\partial x} & h_{1}\frac{\partial r}{\partial y} & h_{1}\frac{\partial r}{\partial z}\\ h_{2}\frac{\partial\theta}{\partial x} & h_{2}\frac{\partial\theta}{\partial y} & h_{2}\frac{\partial\theta}{\partial z}\\ h_{3}\frac{\partial\phi}{\partial x} & h_{3}\frac{\partial\phi}{\partial y} & h_{3}\frac{\partial\phi}{\partial z} \end{array}\right)\left(\begin{array}{c} A_{x}\\ A_{y}\\ A_{z} \end{array}\right)\\ & =\left(\begin{array}{ccc} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta\\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta\\ -\sin\phi & \cos\phi & 0 \end{array}\right)\left(\begin{array}{c} A_{x}\\ A_{y}\\ A_{z} \end{array}\right) \end{align*} \begin{align*} \left(\begin{array}{c} A_{x}\\ A_{y}\\ A_{z} \end{array}\right) & =\left(\begin{array}{ccc} \frac{1}{h_{1}}\frac{\partial x}{\partial r} & \frac{1}{h_{2}}\frac{\partial x}{\partial\theta} & \frac{1}{h_{3}}\frac{\partial x}{\partial\phi}\\ \frac{1}{h_{1}}\frac{\partial y}{\partial r} & \frac{1}{h_{2}}\frac{\partial y}{\partial\theta} & \frac{1}{h_{3}}\frac{\partial y}{\partial\phi}\\ \frac{1}{h_{1}}\frac{\partial z}{\partial r} & \frac{1}{h_{2}}\frac{\partial z}{\partial\theta} & \frac{1}{h_{3}}\frac{\partial z}{\partial\phi} \end{array}\right)\left(\begin{array}{c} A_{r}\\ A_{\theta}\\ A_{\phi} \end{array}\right)\\ & =\left(\begin{array}{ccc} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi\\ \cos\phi & -\sin\theta & 0 \end{array}\right)\left(\begin{array}{c} A_{r}\\ A_{\theta}\\ A_{\phi} \end{array}\right) \end{align*} となる。
また逆行列を使って
\begin{align*} \left(\begin{array}{c} A_{x}\\ A_{y}\\ A_{z} \end{array}\right) & =\left(\begin{array}{ccc} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta\\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta\\ -\sin\phi & \cos\phi & 0 \end{array}\right)^{-1}\left(\begin{array}{c} A_{r}\\ A_{\theta}\\ A_{\phi} \end{array}\right)\\ & =\left(\begin{array}{ccc} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi\\ \cos\phi & -\sin\theta & 0 \end{array}\right)\left(\begin{array}{c} A_{r}\\ A_{\theta}\\ A_{\phi} \end{array}\right) \end{align*} としても求まる。
円柱座標
極座標\[ \begin{cases} x=r\sin\theta\\ y=r\cos\theta\\ z=z \end{cases} \] \[ \begin{cases} r=\sqrt{x^{2}+y^{2}}\\ \theta=\tan^{\bullet}\left(\frac{y}{x}\right)\\ z=z \end{cases} \] のとき、
\[ \left(\begin{array}{ccc} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial\theta} & \frac{\partial x}{\partial\phi}\\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial\theta} & \frac{\partial y}{\partial\phi}\\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial\theta} & \frac{\partial z}{\partial\phi} \end{array}\right)=\left(\begin{array}{ccc} \cos\theta & -r\sin\theta & 0\\ \sin\theta & r\cos\theta & 0\\ 0 & 0 & 1 \end{array}\right) \] \[ \left(\begin{array}{ccc} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} & \frac{\partial r}{\partial z}\\ \frac{\partial\theta}{\partial x} & \frac{\partial\theta}{\partial y} & \frac{\partial\theta}{\partial z}\\ \frac{\partial z}{\partial x} & \frac{\partial z}{\partial y} & \frac{\partial z}{\partial z} \end{array}\right)=\left(\begin{array}{ccc} \cos\theta & \sin\theta & 0\\ -\frac{\sin\theta}{r} & \frac{\cos\theta}{r} & 0\\ 0 & 0 & 1 \end{array}\right) \] \[ \begin{cases} h_{r}=1\\ h_{\theta}=r\\ h_{\phi}=1 \end{cases} \] である。
これより、基底ベクトル同士の関係は
\begin{align*} \left(\begin{array}{c} \boldsymbol{e}_{r}\\ \boldsymbol{e}_{\theta}\\ \boldsymbol{e}_{z} \end{array}\right) & =\left(\begin{array}{ccc} \frac{1}{h_{1}}\frac{\partial x}{\partial r} & \frac{1}{h_{1}}\frac{\partial y}{\partial r} & \frac{1}{h_{1}}\frac{\partial z}{\partial r}\\ \frac{1}{h_{2}}\frac{\partial x}{\partial\theta} & \frac{1}{h_{2}}\frac{\partial y}{\partial\theta} & \frac{1}{h_{2}}\frac{\partial z}{\partial\theta}\\ \frac{1}{h_{3}}\frac{\partial x}{\partial z} & \frac{1}{h_{3}}\frac{\partial y}{\partial z} & \frac{1}{h_{3}}\frac{\partial z}{\partial z} \end{array}\right)\left(\begin{array}{c} \boldsymbol{i}\\ \boldsymbol{j}\\ \boldsymbol{k} \end{array}\right)\\ & =\left(\begin{array}{ccc} \cos\phi & \sin\theta & 0\\ -\sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} \boldsymbol{i}\\ \boldsymbol{j}\\ \boldsymbol{k} \end{array}\right) \end{align*} \begin{align*} \left(\begin{array}{c} \boldsymbol{i}\\ \boldsymbol{j}\\ \boldsymbol{k} \end{array}\right) & =\left(\begin{array}{ccc} h_{1}\frac{\partial r}{\partial x} & h_{2}\frac{\partial\theta}{\partial x} & h_{3}\frac{\partial z}{\partial x}\\ h_{1}\frac{\partial r}{\partial y} & h_{2}\frac{\partial\theta}{\partial y} & h_{3}\frac{\partial z}{\partial y}\\ h_{1}\frac{\partial r}{\partial z} & h_{2}\frac{\partial\theta}{\partial z} & h_{3}\frac{\partial z}{\partial z} \end{array}\right)\left(\begin{array}{c} \boldsymbol{e}_{r}\\ \boldsymbol{e}_{\theta}\\ \boldsymbol{e}_{z} \end{array}\right)\\ & =\left(\begin{array}{ccc} \cos\phi & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} \boldsymbol{e}_{r}\\ \boldsymbol{e}_{\theta}\\ \boldsymbol{e}_{z} \end{array}\right) \end{align*} となる。
ベクトルの成分同士の関係は
\begin{align*} \left(\begin{array}{c} A_{r}\\ A_{\theta}\\ A_{z} \end{array}\right) & =\left(\begin{array}{ccc} h_{1}\frac{\partial r}{\partial x} & h_{1}\frac{\partial r}{\partial y} & h_{1}\frac{\partial r}{\partial z}\\ h_{2}\frac{\partial\theta}{\partial x} & h_{2}\frac{\partial\theta}{\partial y} & h_{2}\frac{\partial\theta}{\partial z}\\ h_{3}\frac{\partial z}{\partial x} & h_{3}\frac{\partial z}{\partial y} & h_{3}\frac{\partial z}{\partial z} \end{array}\right)\left(\begin{array}{c} A_{x}\\ A_{y}\\ A_{z} \end{array}\right)\\ & =\left(\begin{array}{ccc} \cos\theta & \sin\theta & 0\\ -\sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} A_{x}\\ A_{y}\\ A_{z} \end{array}\right) \end{align*} \begin{align*} \left(\begin{array}{c} A_{x}\\ A_{y}\\ A_{z} \end{array}\right) & =\left(\begin{array}{ccc} \frac{1}{h_{1}}\frac{\partial x}{\partial r} & \frac{1}{h_{2}}\frac{\partial x}{\partial\theta} & \frac{1}{h_{3}}\frac{\partial x}{\partial z}\\ \frac{1}{h_{1}}\frac{\partial y}{\partial r} & \frac{1}{h_{2}}\frac{\partial y}{\partial\theta} & \frac{1}{h_{3}}\frac{\partial y}{\partial z}\\ \frac{1}{h_{1}}\frac{\partial z}{\partial r} & \frac{1}{h_{2}}\frac{\partial z}{\partial\theta} & \frac{1}{h_{3}}\frac{\partial z}{\partial z} \end{array}\right)\left(\begin{array}{c} A_{r}\\ A_{\theta}\\ A_{z} \end{array}\right)\\ & =\left(\begin{array}{ccc} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} A_{r}\\ A_{\theta}\\ A_{\phi} \end{array}\right) \end{align*} となる。
(1)
\(\boldsymbol{u}_{k}\)の方向は位置ベクトルを\(q_{k}\)以外を固定して\(q_{k}\)だけ動かしたベクトルで、大きさは\(1\)のベクトルなので、\begin{align*} \boldsymbol{u}_{i} & =\frac{\frac{\partial\boldsymbol{r}}{\partial q_{i}}}{\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{i}}\right\Vert }\\ & =\frac{1}{h_{i}}\frac{\partial\boldsymbol{r}}{\partial q_{i}} \end{align*} また、
\begin{align*} \boldsymbol{e}_{k} & =\frac{\partial\boldsymbol{r}}{\partial x_{k}}\\ & =\sum_{i}\frac{\partial q_{i}}{\partial x_{k}}\frac{\partial\boldsymbol{r}}{\partial q_{i}}\\ & =\sum_{i}\frac{\partial q_{i}}{\partial x_{k}}h_{i}\boldsymbol{u}_{i} \end{align*} となる。
(2)
\begin{align*} \sum_{k}\frac{\partial x_{k}}{\partial q_{i}}\frac{\partial x_{k}}{\partial q_{j}} & =\sum_{k,l}\frac{\partial x_{k}}{\partial q_{i}}\frac{\partial x_{l}}{\partial q_{j}}\boldsymbol{e}_{k}\cdot\boldsymbol{e}_{l}\\ & =\frac{\partial\boldsymbol{r}}{\partial q_{i}}\cdot\frac{\partial\boldsymbol{r}}{\partial q_{j}}\\ & =h_{i}h_{j}\boldsymbol{u}_{i}\cdot\boldsymbol{u}_{j}\\ & =h_{i}h_{j}\delta_{ij} \end{align*}(2)-2
\begin{align*} \delta_{ij} & =\boldsymbol{u}_{i}\cdot\boldsymbol{u}_{j}\\ & =\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{i}}\right\Vert ^{-1}\frac{\partial\boldsymbol{r}}{\partial q_{i}}\cdot\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{j}}\right\Vert ^{-1}\frac{\partial\boldsymbol{r}}{\partial q_{j}}\\ & =\sum_{l.m}\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{i}}\right\Vert ^{-1}\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{j}}\right\Vert ^{-1}\frac{\partial x_{l}}{\partial q_{i}}\frac{\partial x_{m}}{\partial q_{j}}\boldsymbol{e}_{l}\cdot\boldsymbol{e}_{m}\\ & =\sum_{l}\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{i}}\right\Vert ^{-1}\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{j}}\right\Vert ^{-1}\frac{\partial x_{l}}{\partial q_{i}}\frac{\partial x_{l}}{\partial q_{j}}\\ & =\frac{1}{h_{i}h_{j}}\sum_{l}\frac{\partial x_{l}}{\partial q_{i}}\frac{\partial x_{l}}{\partial q_{j}} \end{align*} これより、\[ h_{i}h_{j}\delta_{ij}=\sum_{l}\frac{\partial x_{l}}{\partial q_{i}}\frac{\partial x_{l}}{\partial q_{j}} \]
(3)
\begin{align*} ds^{2} & =\sum_{i}dx_{i}^{2}\\ & =d\boldsymbol{r}\cdot d\boldsymbol{r}\\ & =\sum_{i,j}\frac{\partial\boldsymbol{r}}{\partial q_{i}}\cdot\frac{\partial\boldsymbol{r}}{\partial q_{j}}dq_{i}dq_{j}\\ & =\sum_{i,j,l,m}\frac{\partial x_{l}}{\partial q_{i}}\frac{\partial x_{m}}{\partial q_{j}}\boldsymbol{e}_{l}\cdot\boldsymbol{e}_{m}dq_{i}dq_{j}\\ & =\sum_{i,j,l,m}\frac{\partial x_{l}}{\partial q_{i}}\frac{\partial x_{m}}{\partial q_{j}}\delta_{lm}dq_{i}dq_{j}\\ & =\sum_{i,j,l}\frac{\partial x_{l}}{\partial q_{i}}\frac{\partial x_{l}}{\partial q_{j}}dq_{i}dq_{j}\\ & =\sum_{i,j}\delta_{ij}h_{i}h_{j}dq_{i}dq_{j}\\ & =\sum_{i}h_{i}^{2}dq_{i}^{2} \end{align*}(4)
直交座標での単位基底ベクトルを\(\boldsymbol{e}_{k}\)として、直交曲線座標での単位基底ベクトルを\(\boldsymbol{e}_{k}'\)とすると、であるので、
\begin{align*} \sum_{k}A_{k}'\boldsymbol{e}_{k}' & =\sum_{j}A_{j}\boldsymbol{e}_{j}\\ & =\sum_{j}A_{j}\sum_{k}\frac{\partial q_{k}}{\partial x_{j}}h_{k}\boldsymbol{e}_{k}'\\ & =\sum_{k}\left(\sum_{j}A_{j}\frac{\partial q_{k}}{\partial x_{j}}h_{k}\right)\boldsymbol{e}_{k}' \end{align*} となる。
これより、
\[ A_{k}'=\sum_{j}A_{j}\frac{\partial q_{k}}{\partial x_{j}}h_{k} \] となる。
また、
\begin{align*} \sum_{k}A_{k}\boldsymbol{e}_{k} & =\sum_{j}A_{j}'\boldsymbol{e}_{j}'\\ & =\sum_{j}A_{j}'\frac{1}{h_{j}}\frac{\partial\boldsymbol{r}}{\partial q_{j}}\\ & =\sum_{j}\sum_{k}A_{j}'\frac{1}{h_{j}}\frac{\partial x_{k}}{\partial q_{j}}\boldsymbol{e}_{k}\\ & =\sum_{k}\left(\sum_{j}A_{j}'\frac{1}{h_{j}}\frac{\partial x_{k}}{\partial q_{j}}\right)\boldsymbol{e}_{k} \end{align*} となるので、
\[ A_{k}=\left(\sum_{j}A_{j}'\frac{1}{h_{j}}\frac{\partial x_{k}}{\partial q_{j}}\right) \] となる。
(5)
基底ベクトルを\(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}\)とする直交座標系\(\left(x_{1},x_{2},\cdots,x_{n}\right)\)から基底ベクトルを\(\boldsymbol{u}_{1},\boldsymbol{u}_{2},\cdots,\boldsymbol{u}_{n}\)とする直交曲線座標系\(\left(q_{1},q_{2},\cdots,q_{n}\right)\)へのヤコビ行列を\(J\)とする。このとき、
\begin{align*} \boldsymbol{u}_{i} & =\frac{1}{h_{i}}\frac{\partial\boldsymbol{r}}{\partial q_{i}}\\ & =\frac{1}{h_{i}}\frac{\partial}{\partial q_{i}}\sum_{j}x_{j}\boldsymbol{e}_{j}\\ & =\frac{1}{h_{i}}\sum_{j}\frac{\partial x_{j}}{\partial q_{i}}\boldsymbol{e}_{j} \end{align*} であるので、
\begin{align*} \left(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}\right)J & =\left(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}\right)\left(\begin{array}{cccc} \frac{\partial x_{1}}{\partial q_{1}} & \frac{\partial x_{1}}{\partial q_{2}} & \cdots & \frac{\partial x_{1}}{\partial q_{n}}\\ \frac{\partial x_{2}}{\partial q_{1}} & \frac{\partial x_{2}}{\partial q_{2}} & \cdots & \frac{\partial x_{2}}{\partial q_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial x_{n}}{\partial q_{1}} & \frac{\partial x_{n}}{\partial q_{2}} & \cdots & \frac{\partial x_{n}}{\partial q_{n}} \end{array}\right)\\ & =\left(\sum_{j}\frac{\partial x_{j}}{\partial q_{1}}\boldsymbol{e}_{j},\sum_{j}\frac{\partial x_{j}}{\partial q_{2}}\boldsymbol{e}_{j},\cdots,\sum_{j}\frac{\partial x_{j}}{\partial q_{n}}\boldsymbol{e}_{j}\right)\\ & =\left(h_{1}\boldsymbol{u}_{1},h_{2}\boldsymbol{u}_{2},\cdots,h_{n}\boldsymbol{u}_{n}\right) \end{align*} となる。
これより、
\begin{align*} \left(\det J\right)^{2} & =\det\left(J^{T}J\right)\\ & =\det\left(J^{T}\left(\begin{array}{c} \boldsymbol{e}_{1}\\ \boldsymbol{e}_{2}\\ \vdots\\ \boldsymbol{e}_{n} \end{array}\right)\left(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}\right)J\right)\cmt{\because\left(\begin{array}{c} \boldsymbol{e}_{1}\\ \boldsymbol{e}_{2}\\ \vdots\\ \boldsymbol{e}_{n} \end{array}\right)\left(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}\right)=I_{n}}\\ & =\det\left(\left(\left(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}\right)J\right)^{T}\left(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}\right)J\right)\\ & =\det\left(\left(\left(h_{1}\boldsymbol{u}_{1},h_{2}\boldsymbol{u}_{2},\cdots,h_{n}\boldsymbol{u}_{n}\right)\right)^{T}\left(h_{1}\boldsymbol{u}_{1},h_{2}\boldsymbol{u}_{2},\cdots,h_{n}\boldsymbol{u}_{n}\right)\right)\cmt{\because\left(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}\right)J=\left(h_{1}\boldsymbol{u}_{1},h_{2}\boldsymbol{u}_{2},\cdots,h_{n}\boldsymbol{u}_{n}\right)}\\ & =\det\left(\left(\begin{array}{c} h_{1}\boldsymbol{u}_{1}\\ h_{2}\boldsymbol{u}_{2}\\ \vdots\\ h_{n}\boldsymbol{u}_{n} \end{array}\right)\left(h_{1}\boldsymbol{u}_{1},h_{2}\boldsymbol{u}_{2},\cdots,h_{n}\boldsymbol{u}_{n}\right)\right)\\ & =\det\left(\begin{array}{cccc} h_{1}h_{1}\boldsymbol{u}_{1}\cdot\boldsymbol{u}_{1} & h_{1}h_{2}\boldsymbol{u}_{1}\cdot\boldsymbol{u}_{2} & \cdots & h_{1}h_{n}\boldsymbol{u}_{1}\cdot\boldsymbol{u}_{n}\\ h_{2}h_{1}\boldsymbol{u}_{2}\cdot\boldsymbol{u}_{1} & h_{2}h_{2}\boldsymbol{u}_{2}\cdot\boldsymbol{u}_{2} & \cdots\cdots & h_{2}h_{n}\boldsymbol{u}_{2}\cdot\boldsymbol{u}_{n}\\ \vdots & \vdots & \ddots & \vdots\\ h_{n}h_{1}\boldsymbol{u}_{n}\cdot\boldsymbol{u}_{1} & h_{n}h_{2}\boldsymbol{u}_{n}\cdot\boldsymbol{u}_{2} & \cdots\cdots & h_{n}h_{n}\boldsymbol{u}_{n}\cdot\boldsymbol{u}_{n} \end{array}\right)\\ & =\det\left(\begin{array}{cccc} h_{1}h_{1}\delta_{1,1} & h_{1}h_{2}\delta_{1,2} & \cdots & h_{1}h_{n}\delta_{1,n}\\ h_{2}h_{1}\delta_{2,1} & h_{2}h_{2}\delta_{2,2} & \cdots\cdots & h_{2}h_{n}\delta_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ h_{n}h_{1}\delta_{n,1} & h_{n}h_{2}\delta_{n,2} & \cdots\cdots & h_{n}h_{n}\delta_{n,n} \end{array}\right)\\ & =\det\left(\begin{array}{cccc} h_{1}h_{1} & 0 & \cdots & 0\\ 0 & h_{2}h_{2} & \cdots\cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots\cdots & 0 \end{array}\right)\\ & =\prod_{k=1}^{n}h_{k}^{2}\\ & =\left(\prod_{k=1}^{n}h_{k}\right)^{2} \end{align*} となるので、\(\det J=\pm\prod_{k=1}^{n}h_{k}\)となる。
これより、ヤコビアン\(\det J\)の絶対値は
\begin{align*} \left|\det J\right| & =\left|\pm\prod_{k=1}^{n}h_{k}\right|\\ & =\prod_{k=1}^{n}h_{k} \end{align*} となる。
従って題意は成り立つ。
ページ情報
| タイトル | スケール因子・微小線素と単位基底ベクトル・ベクトルの成分同士の関係 |
| URL | https://www.nomuramath.com/r4mfkumq/ |
| SNSボタン |
直交曲線座標でのナブラ演算子・回転・発散・ラプラシアン
\[
\boldsymbol{\nabla}\cdot\boldsymbol{A}=\frac{1}{h}\sum_{i}\frac{\partial}{\partial q_{i}}\frac{A_{i}h}{h_{i}}
\]
直交曲線座標での単位基底ベクトルの回転・発散
\[
\boldsymbol{\nabla}\cdot\boldsymbol{u}_{i}=\frac{1}{hh_{i}}\frac{\partial}{\partial q_{i}}h-\frac{1}{h_{i}^{2}}\frac{\partial}{\partial q_{i}}h_{i}
\]
直交曲線座標での性質
\[
h_{i}\boldsymbol{\nabla}q_{i}=\frac{1}{h_{i}}\frac{\partial\boldsymbol{r}}{\partial q_{i}}
\]
勾配の方向と方向微分
\[
\nabla_{\boldsymbol{v}}f\left(\boldsymbol{r}\right):=\boldsymbol{v}\cdot\boldsymbol{\nabla}f
\]