座標系 (Coordinate system)

デカルト座標 (Cartesian coordinate system)

極座標 (Polar coordinate system)

x=rcosθy=rsinθ\begin{aligned} x & = r\cos \theta \\ y & = r\sin \theta \end{aligned}

放物線座標 (Parabolic coordinates)

Two-dimensional parabolic coordinates (σ,τ) (\sigma , \tau ) are defined by the equations

x=στy=12(τ2−σ2)\begin{aligned} x & = \sigma \tau \\ y & = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \end{aligned}

The curves of constant σ \sigma form confocal parabolae

2y=x2σ2−σ22y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2}

円柱座標 (Cylindrical coordinate system)

x=ρcosφy=ρsinφz=z\begin{aligned} x & = \rho \cos \varphi \\ y & = \rho \sin \varphi \\ z & = z \end{aligned}
dr=dρeρ+ρdφeφ+dzezd \mathbf{r} = d \rho \mathbf{e_{\rho}} + \rho d \varphi \mathbf{e_{\varphi}} + dz \mathbf{e_{z}}
dV=ρdρdφdzd \mathbf{V} = \rho d \rho d \varphi dz

放物線柱座標 (Parabolic cylindrical coordinates)

x=στy=12(τ2−σ2)z=z\begin{aligned} x & = \sigma \tau \\ y & = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\ z & = z \end{aligned}