Categories: Mathematics, Physics.

# Parabolic cylindrical coordinates

Parabolic cylindrical coordinates extend parabolic coordinates $(\sigma, \tau)$ to 3D, by describing a point in space using the variables $(\sigma, \tau, z)$. The $z$-axis is the same as in the Cartesian system, (hence the name cylindrical), while the coordinate lines of $\sigma$ and $\tau$ are confocal parabolas.

Cartesian coordinates $(x, y, z)$ and this system $(\sigma, \tau, z)$ are related by:

\begin{aligned} \boxed{ \begin{aligned} x &= \frac{1}{2} (\tau^2 - \sigma^2) \\ y &= \sigma \tau \\ z &= z \end{aligned} } \end{aligned}

Conversely, a point given in $(x, y, z)$ can be converted to $(\sigma, \tau, z)$ using these formulae:

\begin{aligned} \boxed{ \begin{aligned} \sigma &= \sqrt{\sqrt{x^2 + y^2} - x} \\ \tau &= \sgn(y) \sqrt{\sqrt{x^2 + y^2} + x} \\ z &= z \end{aligned} } \end{aligned}

Parabolic cylindrical coordinates form an orthogonal curvilinear system, whose scale factors $h_\sigma$, $h_\tau$ and $h_z$ we need. To get those, we calculate the unnormalized local basis:

\begin{aligned} h_\sigma \vu{e}_\sigma &= \vu{e}_x \pdv{x}{\sigma} + \vu{e}_y \pdv{y}{\sigma} + \vu{e}_z \pdv{z}{\sigma} \\ &= - \vu{e}_x \sigma + \vu{e}_y \tau \\ h_\tau \vu{e}_\tau &= \vu{e}_x \pdv{x}{\tau} + \vu{e}_y \pdv{y}{\tau} + \vu{e}_z \pdv{z}{\tau} \\ &= \vu{e}_x \tau + \vu{e}_y \sigma \\ h_\sigma \vu{e}_\sigma &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z} \\ &= \vu{e}_z \end{aligned}

By normalizing the local basis vectors $\vu{e}_\sigma$, $\vu{e}_\tau$ and $\vu{e}_z$, we arrive at these expressions, where we have defined the abbreviation $\rho$ for convenience:

\begin{aligned} \boxed{ \begin{aligned} h_\sigma &= \rho \equiv \sqrt{\sigma^2 + \tau^2} \\ h_\tau &= \rho \equiv \sqrt{\sigma^2 + \tau^2} \\ h_z &= 1 \end{aligned} } \qquad\qquad \boxed{ \begin{aligned} \vu{e}_\sigma &= - \vu{e}_x \frac{\sigma}{\rho} + \vu{e}_y \frac{\tau}{\rho} \\ \vu{e}_\tau &= \vu{e}_x \frac{\tau}{\rho} + \vu{e}_y \frac{\sigma}{\rho} \\ \vu{e}_z &= \vu{e}_z \end{aligned} } \end{aligned}

Thanks to these scale factors, we can easily convert calculus from the Cartesian system using the standard formulae for orthogonal curvilinear coordinates.

## Differential elements

For line integrals, the tangent vector element $\dd{\vb{\ell}}$ for a curve is as follows:

\begin{aligned} \boxed{ \dd{\vb{\ell}} = \vu{e}_\sigma \: \rho \dd{\sigma} + \: \vu{e}_\tau \: \rho \dd{\tau} + \: \vu{e}_z \dd{z} } \end{aligned}

For surface integrals, the normal vector element $\dd{\vb{S}}$ for a surface is given by:

\begin{aligned} \boxed{ \dd{\vb{S}} = \vu{e}_\sigma \: \rho \dd{\tau} \dd{z} + \: \vu{e}_\tau \: \rho \dd{\sigma} \dd{z} + \: \vu{e}_z \: \rho^2 \dd{\sigma} \dd{\tau} } \end{aligned}

And for volume integrals, the infinitesimal volume $\dd{V}$ takes the following form:

\begin{aligned} \boxed{ \dd{V} = \rho^2 \dd{\sigma} \dd{\tau} \dd{z} } \end{aligned}

## Common operations

The basic vector operations (gradient, divergence, curl and Laplacian) are given by:

\begin{aligned} \boxed{ \nabla f = \vu{e}_\sigma \frac{1}{\rho} \pdv{f}{\sigma} + \vu{e}_\tau \frac{1}{\rho} \pdv{f}{\tau} + \vu{e}_z \pdv{f}{z} } \end{aligned} \begin{aligned} \boxed{ \nabla \cdot \vb{V} = \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\sigma V_\sigma}{\rho^3} + \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\tau V_\tau}{\rho^3} + \pdv{V_z}{z} } \end{aligned} \begin{aligned} \boxed{ \begin{aligned} \nabla \times \vb{V} &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \bigg) \\ &\quad\: + \vu{e}_\tau \bigg( \pdv{V_\sigma}{z} - \frac{1}{\rho} \pdv{V_z}{\sigma} \bigg) \\ &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} + \frac{\sigma V_\tau}{\rho^3} - \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\tau V_\sigma}{\rho^3} \bigg) \end{aligned} } \end{aligned} \begin{aligned} \boxed{ \nabla^2 f = \frac{1}{\rho^2} \pdvn{2}{f}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{f}{\tau} + \pdvn{2}{f}{z} } \end{aligned}

## Uncommon operations

Uncommon operations include: the gradient of a divergence $\nabla (\nabla \cdot \vb{V})$, the gradient of a vector $\nabla \vb{V}$, the advection of a vector $(\vb{U} \cdot \nabla) \vb{V}$ with respect to $\vb{U}$, the Laplacian of a vector $\nabla^2 \vb{V}$, and the divergence of a 2nd-order tensor $\nabla \cdot \overline{\overline{\vb{T}}}$:

\begin{aligned} \boxed{ \begin{aligned} \nabla (\nabla \cdot \vb{V}) &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \mpdv{V_\tau}{\sigma}{\tau} + \frac{1}{\rho} \mpdv{V_z}{\sigma}{z} \\ &\qquad\qquad + \frac{\tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{\sigma}{\rho^4} \pdv{V_\tau}{\tau} + \frac{\rho^2 - 3 \sigma^2}{\rho^6} V_\sigma - \frac{3 \sigma \tau V_\tau}{\rho^6} \bigg) \\ &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \mpdv{V_\sigma}{\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \frac{1}{\rho} \mpdv{V_z}{\tau}{z} \\ &\qquad\qquad - \frac{\tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{\sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{3 \sigma \tau V_\sigma}{\rho^6} + \frac{\rho^2 - 3 \tau^2}{\rho^6} V_\tau \bigg) \\ &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \mpdv{V_\sigma}{z}{\sigma} + \frac{1}{\rho} \mpdv{V_\tau}{z}{\tau} + \pdvn{2}{V_z}{z} + \frac{\sigma}{\rho^3} \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} \pdv{V_\tau}{z} \bigg) \end{aligned} } \end{aligned} \begin{aligned} \boxed{ \begin{aligned} \nabla \vb{V} &= \quad \vu{e}_\sigma \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\tau V_\tau}{\rho^3} \bigg) + \vu{e}_\sigma \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} - \frac{\tau V_\sigma}{\rho^3} \bigg) + \vu{e}_\sigma \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\sigma} \\ &\quad\: + \vu{e}_\tau \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\sigma V_\tau}{\rho^3} \bigg) + \vu{e}_\tau \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\sigma V_\sigma}{\rho^3} \bigg) + \vu{e}_\tau \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\tau} \\ &\quad\: + \vu{e}_z \vu{e}_\sigma \pdv{V_\sigma}{z} + \vu{e}_z \vu{e}_\tau \pdv{V_\tau}{z} + \vu{e}_z \vu{e}_z \pdv{V_z}{z} \end{aligned} } \end{aligned} \begin{aligned} \boxed{ \begin{aligned} (\vb{U} \cdot \nabla) \vb{V} &= \quad \vu{e}_\sigma \bigg( \frac{U_\sigma}{\rho} \pdv{V_\sigma}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\sigma}{\tau} + U_z \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} U_\sigma V_\tau - \frac{\sigma}{\rho^3} U_\tau V_\tau \bigg) \\ &\quad\: + \vu{e}_\tau \bigg( \frac{U_\sigma}{\rho} \pdv{V_\tau}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\tau}{\tau} + U_z \pdv{V_\tau}{z} + \frac{\sigma}{\rho^3} U_\tau V_\sigma - \frac{\tau}{\rho^3} U_\sigma V_\sigma \bigg) \\ &\quad\: + \vu{e}_z \bigg( \frac{U_\sigma}{\rho} \pdv{V_z}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_z}{\tau} + U_z \pdv{V_z}{z} \bigg) \end{aligned} } \end{aligned} \begin{aligned} \boxed{ \begin{aligned} \nabla^2 \vb{V} &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\tau} + \pdvn{2}{V_\sigma}{z} + \frac{2 \tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{2 \sigma}{\rho^4} \pdv{V_\tau}{\tau} - \frac{V_\sigma}{\rho^4} \bigg) \\ &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \pdvn{2}{V_\tau}{z} - \frac{2 \tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{2 \sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{V_\tau}{\rho^4} \bigg) \\ &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho^2} \pdvn{2}{V_z}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_z}{\tau} + \pdvn{2}{V_z}{z} \bigg) \end{aligned} } \end{aligned} \begin{aligned} \boxed{ \begin{aligned} \nabla \cdot \overline{\overline{\mathbf{T}}} &= \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{T_{\sigma \sigma}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \sigma}}{\tau} + \pdv{T_{z \sigma}}{z} + \frac{\sigma T_{\sigma \sigma}}{\rho^3} + \frac{\tau T_{\sigma \tau}}{\rho^3} + \frac{\tau T_{\tau \sigma}}{\rho^3} - \frac{\sigma T_{\tau \tau}}{\rho^3} \bigg) \\ &+ \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{T_{\sigma \tau}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \tau}}{\tau} + \pdv{T_{k \tau}}{z} - \frac{\tau T_{\sigma \sigma}}{\rho^3} + \frac{\sigma T_{\sigma \tau}}{\rho^3} + \frac{\sigma T_{\tau \sigma}}{\rho^3} + \frac{\tau T_{\tau \tau}}{\rho^3} \bigg) \\ &+ \vu{e}_z \bigg( \frac{1}{\rho} \pdv{T_{\sigma z}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau z}}{\tau} + \pdv{T_{zz}}{z} + \frac{\sigma T_{\sigma z}}{\rho^3} + \frac{\tau T_{\tau z}}{\rho^3} \bigg) \end{aligned} } \end{aligned}

## References

1. M.L. Boas, Mathematical methods in the physical sciences, 2nd edition, Wiley.