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

by nomura · 2025年6月24日

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。

(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。

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。

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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
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SNSボタン

アインシュタインの和の既約

勾配の方向と方向微分

\[ \nabla_{\boldsymbol{v}}f\left(\boldsymbol{r}\right):=\boldsymbol{v}\cdot\boldsymbol{\nabla}f \]

ナブラ演算子・勾配・発散・回転・ラプラシアンの定義

\[ \boldsymbol{\nabla}:=\boldsymbol{e}_{i}\partial_{i} \]

直交曲線座標での単位基底ベクトルの回転・発散

\[ \boldsymbol{\nabla}\cdot\boldsymbol{u}_{i}=\frac{1}{hh_{i}}\frac{\partial}{\partial q_{i}}h-\frac{1}{h_{i}^{2}}\frac{\partial}{\partial q_{i}}h_{i} \]