直交曲線座標系における回転 (rot)

回転の定義

ベクトル場 \(\mathbf f\) の回転の定義は次の式で与えられます。
\[\begin{align}
\nabla \times \mathbf f
=
\left(
\mathbf e_x \frac{\partial}{\partial x} +
\mathbf e_y \frac{\partial}{\partial y} +
\mathbf e_z \frac{\partial}{\partial z}
\right)
\times
\left(
f_x \mathbf e_x +
f_y \mathbf e_y +
f_z \mathbf e_z
\right)
\end{align}\]
ただし \(f_x = \mathbf f \cdot \mathbf e_x\)、\(f_y = \mathbf f \cdot \mathbf e_y\)、\(f_z = \mathbf f \cdot \mathbf e_z\) です。

直交曲線座標への変換

直交曲線座標へ変換すると、
\[\begin{align}
\nabla \times \mathbf f
=
&\left[
\mathbf e_x
\left(
\frac{\partial q_1}{\partial x} \frac{\partial }{\partial q_1} +
\frac{\partial q_2}{\partial x} \frac{\partial }{\partial q_2} +
\frac{\partial q_3}{\partial x} \frac{\partial }{\partial q_3}
\right) \right. \nonumber \\
& + \mathbf e_y
\left(
\frac{\partial q_1}{\partial y} \frac{\partial }{\partial q_1} +
\frac{\partial q_2}{\partial y} \frac{\partial }{\partial q_2} +
\frac{\partial q_3}{\partial y} \frac{\partial }{\partial q_3}
\right) \nonumber \\
& +
\left. \mathbf e_z
\left(
\frac{\partial q_1}{\partial y} \frac{\partial}{\partial q_1} +
\frac{\partial q_2}{\partial y} \frac{\partial}{\partial q_2} +
\frac{\partial q_3}{\partial y} \frac{\partial}{\partial q_3}
\right) \right]\times \mathbf f
\end{align}\]
直交曲線座標とデカルト座標の微分変数の交換で証明した \(\displaystyle \frac{\partial q_i}{\partial x} = \frac{1}{h_i^2} \frac{\partial x}{\partial q_i} \) などの関係式を代入して整理すると
\[\begin{align}
\nabla \times \mathbf f
=
&\left[
\frac{1}{h^2_1}
\left(
\mathbf e_x \frac{\partial x}{\partial q_1} +
\mathbf e_y \frac{\partial y}{\partial q_1} +
\mathbf e_z \frac{\partial z}{\partial q_1}
\right) \frac{\partial }{\partial q_1}\right. \nonumber \\
& + \frac{1}{h^2_2}
\left(
\mathbf e_x \frac{\partial x}{\partial q_2} +
\mathbf e_y \frac{\partial y}{\partial q_2} +
\mathbf e_z \frac{\partial z}{\partial q_2}
\right) \frac{\partial }{\partial q_2} \nonumber \\
& + \left.
\frac{1}{h^3_2}
\left(
\mathbf e_x \frac{\partial x}{\partial q_3} +
\mathbf e_y \frac{\partial y}{\partial q_3} +
\mathbf e_z \frac{\partial z}{\partial q_3}
\right) \frac{\partial }{\partial q_3}
\right] \nonumber \\
&\qquad \times
\left(
f_1 \mathbf e_1 +
f_2 \mathbf e_2 +
f_3 \mathbf e_3
\right)
\end{align}\]

ここで、直交曲線座標系における単位ベクトルで説明した \(\mathbf e_i\) の定義
\[
\mathbf e_i =
\frac{1}{h_i}\left(
\mathbf e_x \frac{\partial x}{\partial q_i} +
\mathbf e_y \frac{\partial y}{\partial q_i} +
\mathbf e_z \frac{\partial z}{\partial q_i}
\right)
\]
と見比べると以下の等式を得ます。
\[\begin{align}
\nabla \times \mathbf f
&=
\left(
\frac{\mathbf e_1}{h_1} \frac{\partial }{\partial q_1} +
\frac{\mathbf e_2}{h_2} \frac{\partial }{\partial q_2} +
\frac{\mathbf e_3}{h_3} \frac{\partial }{\partial q_3}
\right)
\times
\left(
f_1 \mathbf e_1 +
f_2 \mathbf e_2 +
f_3 \mathbf e_3
\right) \label{eq:define-curvilinear-rot}
\end{align}\]
これが直交曲線座標系における回転の定義です。

回転の計算

式\eqref{eq:define-curvilinear-rot} の項を一つずつ計算していきます。
\[\begin{align}
\frac{\mathbf e_1}{h_1} \frac{\partial }{\partial q_1} \times (f_1 \mathbf e_1)
&=
\frac{1}{h_1} \frac{\partial f_1}{\partial q_1} \mathbf e_1 \times \mathbf e_1
+ \frac{f_1}{h_1} \mathbf e_1 \times \frac{\partial \mathbf e_1}{\partial q_1} \nonumber\\
&= \frac{f_1}{h_1} \mathbf e_1 \times
\left\{
– \frac{1}{h_2}\frac{\partial h_1}{\partial q_2} \mathbf e_2
– \frac{1}{h_3}\frac{\partial h_1}{\partial q_3} \mathbf e_3
\right\} \nonumber \\
&= -\frac{f_1}{h_1 h_2} \frac{\partial h_1}{\partial q_2} \mathbf e_1 \times \mathbf e_2
-\frac{f_1}{h_1 h_3} \frac{\partial h_1}{\partial q_3} \mathbf e_1 \times \mathbf e_3
\end{align}\]
ここで同じベクトルの外積は 0、すなわち \(\mathbf e_i \times \mathbf e_i = 0\) の関係を使用しています。

第2項は
\[\begin{align}
\frac{\mathbf e_2}{h_2} \frac{\partial }{\partial q_2} \times (f_1 \mathbf e_1)
&=
\frac{1}{h_2} \frac{\partial f_1}{\partial q_2} \mathbf e_2 \times \mathbf e_1 +
\frac{f_1}{h_2} \mathbf e_2 \times \frac{\partial \mathbf e_1}{\partial q_2} \nonumber\\
&=
\frac{1}{h_2} \frac{\partial f_1}{\partial q_2} \mathbf e_2 \times \mathbf e_1 +
\frac{f_1}{h_2} \mathbf e_2 \times
\left\{
\frac{1}{h_1}\frac{\partial h_2}{\partial q_1} \mathbf e_2
\right\} \nonumber \\
&= \frac{1}{h_2} \frac{\partial f_1}{\partial q_1} \mathbf e_2 \times \mathbf e_1
\end{align}\]
同様に第3項は
\[\begin{align}
\frac{\mathbf e_3}{h_3} \frac{\partial }{\partial q_3} \times (f_1 \mathbf e_1)
&= \frac{1}{h_3} \frac{\partial f_1}{\partial q_3} \mathbf e_3 \times \mathbf e_1
\end{align}\]
ゆえに第1項から第3項の合計は以下のように表せます。
\[\begin{align}
\nabla \times (f_1 \mathbf e_1)
=
&
-\frac{1}{h_1 h_2}
\left(f_1 \frac{\partial h_1}{\partial q_2} + h_1 \frac{\partial f_1}{\partial q_2} \right)
\mathbf e_1 \times \mathbf e_2 \nonumber\\
&
+\frac{1}{h_1 h_3}
\left(f_1\frac{\partial h_1}{\partial q_3} + h_1 \frac{\partial f_1}{\partial q_3} \right)
\mathbf e_3 \times \mathbf e_1
\end{align}\]

同様に \(\nabla \times (f_2 \mathbf e_2)\) と \(\nabla \times (f_3 \mathbf e_3)\) を計算して、総和をとると
\[\begin{align}
\nabla \times \mathbf f
=&
\frac{1}{h_1 h_2}
\left\{
\left(f_2 \frac{\partial h_2}{\partial q_1} – f_1 \frac{\partial h_1}{\partial q_2} +
h_2 \frac{\partial f_2}{\partial q_1} – h_1 \frac{\partial f_1}{\partial q_2} \right)
\mathbf e_1 \times \mathbf e_2
\right\} + \nonumber\\
& \frac{1}{h_2 h_3}
\left\{
\left(f_3 \frac{\partial h_3}{\partial q_2} – f_2 \frac{\partial h_3}{\partial q_3} +
h_3 \frac{\partial f_3}{\partial q_2} – h_2 \frac{\partial f_3}{\partial q_3} \right)
\mathbf e_2 \times \mathbf e_3
\right\} + \nonumber\\
& \frac{1}{h_3 h_1}
\left\{
\left(f_1 \frac{\partial h_1}{\partial q_3} – f_3 \frac{\partial h_1}{\partial q_1} +
h_1 \frac{\partial f_1}{\partial q_3} – h_3 \frac{\partial f_1}{\partial q_1} \right)
\mathbf e_3 \times \mathbf e_1
\right\}
\end{align}\]
積関数の微分公式 \( (fg)^\prime = f^\prime g + fg^\prime \) を適用して、
\[\begin{align}
\nabla \times \mathbf f
=&
\frac{\mathbf e_3}{h_1 h_2}
\left(\frac{\partial (f_2 h_2)}{\partial q_1} – \frac{\partial (f_1 h_1)}{\partial q_2} \right) + \nonumber\\
& \frac{\mathbf e_1}{h_2 h_3}
\left(\frac{\partial (f_3 h_3)}{\partial q_2} – \frac{\partial (f_2 h_2)}{\partial q_3} \right) + \nonumber\\
& \frac{\mathbf e_2}{h_3 h_1}
\left(\frac{\partial (f_1 h_1)}{\partial q_3} – \frac{\partial (f_3 h_3)}{\partial q_1} \right) \end{align}\]
ただし、\((\mathbf e_x, \mathbf e_y, \mathbf e_z)\) と \((\mathbf e_1, \mathbf e_2, \mathbf e_3)\) の右手系・左手系が一致すると仮定して、\(\mathbf e_1 \times \mathbf e_2 = \mathbf e_3 \) などを用いています。

直交曲線座標系におけるラプラシアン
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