日: 2014年4月16日

Special curvilinear coordinates

1. Cylindrical coordinates (\rho, \phi, z)

Transformation equations: x = \rho\cos\phi ,\ y = \rho\sin\phi ,\ z = z

where \rho \ge 0 , 0 \le \phi \le 2\pi, -\infty < z < \infty [/latex]. <em>Scale factors</em>: [latex]h_1 = 1,\ h_2 = 1,\ h_3 = 1

Element of arc length: ds^2 = d\rho^2 + \rho^2 d\phi^2 + dz^2

Jacobian: \displaystyle \frac{\partial(x, y, z)}{\partial(\rho, \phi, z)} = \rho

Element of volume: dV = \rho d\rho d\phi dz

Laplacian: \displaystyle \nabla^2U = \frac{1}{\rho}\frac{\partial}{\partial\rho}\left( \rho\frac{\partial U}{\partial\rho} \right) + \frac{1}{\rho^2}\frac{\partial^2U}{\partial\phi^2} + \frac{\partial^2U}{\partial z^2} = \frac{\partial^2U}{\partial\rho^2} + \frac{1}{\rho}\frac{\partial U}{\partial\rho} + \frac{1}{\rho^2}\frac{\partial^2U}{\partial\phi^2} + \frac{\partial^2U}{\partial z^2}

Note that corresponding results can be obtained for polar coordinates in the plane by omitting z dependence. In such case for example, ds^2 = d\rho^2 + \rho^2d\phi^2, while the element of volume is replaced by the element of area, dA = \rho d\rho d\phi.

2. Spherical coordinates (r, \theta, \phi)

Transformation equations: x = r\sin\theta,\ y = r\sin\theta\sin\phi,\ z = r\cos\theta

where r \ge 0,\ 0 \le \theta \le \pi,\ 0 \le \phi \le 2\pi .

Scale factors: h_1 = 1,\ h_2 = r,\ h_3 = r\sin\theta

Element of arc length: ds^2 = dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2

Jacobian: \displaystyle \frac{\partial(x, y, z)}{\partial(r, \theta, \phi)} = r^2\sin\theta

Element of volume: dV = r^2\sin\theta drd\theta d\phi

Laplacian: \displaystyle \nabla^2U = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial U}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial U}{\partial\theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2U}{\partial\phi^2}

Other types of coordinate systems are possible.

特殊な曲線座標

1. 円柱座標系 (\rho, \phi, z)

変換式: x = \rho\cos\phi ,\ y = \rho\sin\phi ,\ z = z

ここで \rho \ge 0 , 0 \le \phi \le 2\pi,

弧長要素: ds^2 = d\rho^2 + \rho^2 d\phi^2 + dz^2

ヤコビアン: \displaystyle \frac{\partial(x, y, z)}{\partial(\rho, \phi, z)} = \rho

体積要素: dV = \rho d\rho d\phi dz

ラプラシアン: \displaystyle \nabla^2U = \frac{1}{\rho}\frac{\partial}{\partial\rho}\left( \rho\frac{\partial U}{\partial\rho} \right) + \frac{1}{\rho^2}\frac{\partial^2U}{\partial\phi^2} + \frac{\partial^2U}{\partial z^2} = \frac{\partial^2U}{\partial\rho^2} + \frac{1}{\rho}\frac{\partial U}{\partial\rho} + \frac{1}{\rho^2}\frac{\partial^2U}{\partial\phi^2} + \frac{\partial^2U}{\partial z^2}

　z 依存性を省略した平面内での極座標系で対応する結果が得られることに注意してください．そのような場合，例えば ds^2 = d\rho^2 + \rho^2d\phi^2 ここで体積要素は面積要素 dA = \rho d\rho d\phi で置換されます．

2. 球面座標系 (r, \theta, \phi)

変換式: x = r\sin\theta,\ y = r\sin\theta\sin\phi,\ z = r\cos\theta

ここで r \ge 0,\ 0 \le \theta \le \pi,\ 0 \le \phi \le 2\pi .

スケール因子: h_1 = 1,\ h_2 = r,\ h_3 = r\sin\theta

弧長要素: ds^2 = dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2

ヤコビアン: \displaystyle \frac{\partial(x, y, z)}{\partial(r, \theta, \phi)} = r^2\sin\theta

体積要素: dV = r^2\sin\theta drd\theta d\phi

ラプラシアン: \displaystyle \nabla^2U = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial U}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial U}{\partial\theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2U}{\partial\phi^2}

他の種類の座標系も可能です．