2 勾配・発散・回転

ここで、計算するカーテシアン座標系と円柱座標系、極座標系の勾配と発散、回転を示し ておく。

2.1 曲線座標の一般形

曲線座標形の微分演算子(勾配、発散、回転)は、スケール因子 $(h_1,h_2,h_3)$を用いて、 以下のように表すことができる。

$$\begin{align*}\begin{aligned}\nabla f= \frac{\partial f}{h_1\partial u_1}\hat{\b... ...rac{\partial f}{h_3\partial u_3}\hat{\boldsymbol{u}}_3 \end{aligned}\end{align*}$$ (1)

$$\begin{align*}\begin{aligned}\div{\boldsymbol{A}} =\frac{1}{h_1h_2h_3}\left[ \fr... ...h_1)+ \frac{\partial}{\partial u_3}(A_3h_1h_2) \right] \end{aligned}\end{align*}$$ (2)

$$\begin{align*}\begin{aligned}\nabla\times \boldsymbol{A}=\frac{1}{h_1h_2h_3} \be... ...partial u_3} \\ A_1h_1 & A_2h_2 & A_3h_3 \end{vmatrix} \end{aligned}\end{align*}$$ (3)

2.2 カーテシアン座標系

$$\begin{align*}\begin{aligned}\nabla f =\frac{\partial f}{\partial x}\boldsymbol{... ...symbol{j}+ \frac{\partial f}{\partial z}\boldsymbol{k} \end{aligned}\end{align*}$$ (4)

$$\begin{align*}\begin{aligned}\div{A} &= \frac{\partial A_x}{\partial x}+ \frac{\partial A_y}{\partial y}+ \frac{\partial A_z}{\partial z} \end{aligned}\end{align*}$$ (5)

$$\begin{align*}\begin{aligned}\nabla\times A &=\left( \if 11 \frac{\partial A_z}{... ...tial^{1} A_x}{\partial y^{1}}\fi \right)\boldsymbol{k} \end{aligned}\end{align*}$$ (6)

2.3 円柱座標系

$$\begin{align*}\begin{aligned}\nabla f &=\frac{\partial f}{\partial r}\hat{\bolds... ...a}}+ \frac{\partial f}{\partial z}\hat{\boldsymbol{z}} \end{aligned}\end{align*}$$ (7)

$$\begin{align*}\begin{aligned}\div{A} &=\frac{1}{r}\left[ \frac{\partial}{\partia... ...tial \theta}+ r\frac{\partial A_z}{\partial z} \right] \end{aligned}\end{align*}$$ (8)

$$\begin{align*}\begin{aligned}\nabla\times A &= \left[\frac{1}{r} \if 11 \frac{\p... ...}{\partial \theta^{1}}\fi \right] \hat{\boldsymbol{z}} \end{aligned}\end{align*}$$ (9)

2.4 極座標系

$$\begin{align*}\begin{aligned}\nabla f &=\frac{\partial f}{\partial f}\hat{\bolds... ...\partial f}{\partial\varphi}\hat{\boldsymbol{\varphi}} \end{aligned}\end{align*}$$ (10)

$$\begin{align*}\begin{aligned}\div{A} &=\frac{1}{r^2}\frac{\partial}{\partial r}(... ...\sin\theta}\frac{\partial A_\varphi}{\partial \varphi} \end{aligned}\end{align*}$$ (11)

$$\begin{align*}\begin{aligned}\nabla\times A= \frac{1}{r\sin\theta} \left[ \if 11... ...rtial \theta^{1}}\fi \right]\hat{\boldsymbol{\varphi}} \end{aligned}\end{align*}$$ (12)

ホームページ: Yamamoto's laboratory
著者: 山本昌志