---
title: "Polar cylindrical coordinates"
sort_title: "Polar cylindrical coordinates"
date: 2021-07-26
categories:
- Mathematics
- Physics
layout: "concept"
---

**Polar cylindrical coordinates** extend polar coordinates $$(r, \varphi)$$ to 3D,
by describing the location of a point in space
using the variables $$(r, \varphi, z)$$.
The $$z$$-axis is unchanged from the Cartesian system,
hence the name *cylindrical*.

[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$
and the cylindrical system $$(r, \varphi, z)$$ are related by:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            x
            &= r \cos{\varphi}
            \\
            y
            &= r \sin{\varphi}
            \\
            z
            &= z
        \end{aligned}
    }
\end{aligned}$$

Conversely, a point given in $$(x, y, z)$$
can be converted to $$(r, \varphi, z)$$ using these formulae,
where $$\mathtt{atan2}$$ is the 2-argument arctangent,
which is needed to handle the signs correctly:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            r
            &= \sqrt{x^2 + y^2}
            \\
            \varphi
            &= \mathtt{atan2}(y, x)
            \\
            z
            &= z
        \end{aligned}
    }
\end{aligned}$$

Polar cylindrical coordinates form
an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/),
whose **scale factors** $$h_r$$, $$h_\varphi$$ and $$h_z$$ we need.
To get those, we calculate the unnormalized local basis:

$$\begin{aligned}
    h_r \vu{e}_r
    &= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r}
    \\
    &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi}
    \\
    h_\varphi \vu{e}_\varphi
    &= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi}
    \\
    &= - \vu{e}_x \: r \sin{\varphi} + \vu{e}_y \: r \cos{\varphi}
    \\
    h_z \vu{e}_z
    &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z}
    \\
    &= \vu{e}_z
\end{aligned}$$

By normalizing the **local basis vectors**
$$\vu{e}_r$$, $$\vu{e}_\varphi$$ and $$\vu{e}_z$$,
we arrive at these expressions:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            h_r
            &= 1
            \\
            h_\varphi
            &= r
            \\
            h_z
            &= 1
        \end{aligned}
    }
    \qquad\qquad
    \boxed{
        \begin{aligned}
            \vu{e}_r
            &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi}
            \\
            \vu{e}_\varphi
            &= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi}
            \\
            \vu{e}_z
            &= \vu{e}_z
        \end{aligned}
    }
\end{aligned}$$

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:

$$\begin{aligned}
    \boxed{
        \dd{\vb{\ell}}
        = \vu{e}_r \dd{r}
        + \: \vu{e}_\varphi \: r \dd{\varphi}
        + \: \vu{e}_z \dd{z}
    }
\end{aligned}$$

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

$$\begin{aligned}
    \boxed{
        \dd{\vb{S}}
        = \vu{e}_r \: r \dd{\varphi} \dd{z}
        + \: \vu{e}_\varphi \dd{r} \dd{z}
        + \: \vu{e}_z \: r \dd{r} \dd{\varphi}
    }
\end{aligned}$$

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

$$\begin{aligned}
    \boxed{
        \dd{V}
        = r \dd{r} \dd{\varphi} \dd{z}
    }
\end{aligned}$$



## Common operations

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

$$\begin{aligned}
    \boxed{
        \nabla f
        = \vu{e}_r \pdv{f}{r}
        + \vu{e}_\varphi \frac{1}{r} \pdv{f}{\varphi}
        + \mathbf{e}_z \pdv{f}{z}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \nabla \cdot \vb{V}
        = \pdv{V_r}{r} + \frac{V_r}{r}
        + \frac{1}{r} \pdv{V_\varphi}{\varphi}
        + \pdv{V_z}{z}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla \times \vb{V}
            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( \pdv{V_r}{z} - \pdv{V_z}{r} \bigg)
            \\
            &\quad\: + \vu{e}_z \bigg( \pdv{V_\varphi}{r} + \frac{V_\varphi}{r} - \frac{1}{r} \pdv{V_r}{\varphi} \bigg)
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \nabla^2 f
        = \pdvn{2}{f}{r} + \frac{1}{r} \pdv{f}{r}
        + \frac{1}{r^2} \pdvn{2}{f}{\varphi}
        + \pdvn{2}{f}{z}
    }
\end{aligned}$$



## 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 $$\nabla \cdot \overline{\overline{\vb{T}}}$$:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla (\nabla \cdot \vb{V})
            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\varphi}{r}{\varphi} + \mpdv{V_z}{r}{z}
            + \frac{1}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi}
            + \frac{1}{r} \mpdv{V_z}{\varphi}{z} + \frac{1}{r^2} \pdv{V_r}{\varphi} \bigg)
            \\
            &\quad\: + \vu{e}_z \bigg( \mpdv{V_r}{z}{r} + \frac{1}{r} \mpdv{V_\varphi}{z}{\varphi} + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_r}{z} \bigg)
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla \vb{V}
            &= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r}
            + \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r}
            + \vu{e}_r \vu{e}_z \pdv{V_z}{r}
            \\
            &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
            + \vu{e}_\varphi \vu{e}_\varphi \bigg( \frac{1}{r} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} \bigg)
            + \vu{e}_\varphi \vu{e}_z \frac{1}{r} \pdv{V_z}{\varphi}
            \\
            &\quad\: + \vu{e}_z \vu{e}_r \pdv{V_r}{z}
            + \vu{e}_z \vu{e}_\varphi \pdv{V_\varphi}{z}
            + \vu{e}_z \vu{e}_z \pdv{V_z}{z}
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            (\vb{U} \cdot \nabla) \vb{V}
            &= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\varphi}{r} \pdv{V_r}{\varphi} + U_z \pdv{V_r}{z}
            - \frac{U_\varphi V_\varphi}{r} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\varphi}{r} \pdv{V_\varphi}{\varphi} + U_z \pdv{V_\varphi}{z}
            + \frac{U_\varphi V_r}{r} \bigg)
            \\
            &\quad\: + \vu{e}_z \bigg( U_r \pdv{V_z}{r} + \frac{U_\varphi}{r} \pdv{V_z}{\varphi} + U_z \pdv{V_z}{z} \bigg)
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla^2 \vb{V}
            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} + \pdvn{2}{V_r}{z}
            + \frac{1}{r} \pdv{V_r}{r} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} + \pdvn{2}{V_\varphi}{z}
            + \frac{2}{r^2} \pdv{V_r}{\varphi} + \frac{1}{r} \pdv{V_\varphi}{r} - \frac{V_\varphi}{r^2} \bigg)
            \\
            &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi}
            + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_z}{r} \bigg)
        \end{aligned}
    }
\end{aligned}$$

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \nabla \cdot \overline{\overline{\mathbf{T}}}
            &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\varphi r}}{\varphi} + \pdv{T_{zr}}{z}
            + \frac{T_{rr}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
            \\
            &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\varphi \varphi}}{\varphi} + \pdv{T_{z \varphi}}{z}
            + \frac{T_{r \varphi}}{r} + \frac{T_{\varphi r}}{r} \bigg)
            \\
            &\quad\: + \vu{e}_z \bigg( \pdv{T_{rz}}{r} + \frac{1}{r} \pdv{T_{\varphi z}}{\varphi} + \pdv{T_{zz}}{z}
            + \frac{T_{rz}}{r} \bigg)
        \end{aligned}
    }
\end{aligned}$$



## References
1.  M.L. Boas,
    *Mathematical methods in the physical sciences*, 2nd edition,
    Wiley.
2.  B. Lautrup,
    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
    CRC Press.