---
title: "Parabolic cylindrical coordinates"
sort_title: "Parabolic cylindrical coordinates"
date: 2021-03-04
categories:
- Mathematics
- Physics
layout: "concept"
---

**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](/know/concept/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](/know/concept/orthogonal-curvilinear-coordinates/),
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.