Special curvilinear coordinates

Contents hide

1 1. Cylindrical coordinates
2 2. Spherical 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.