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---
title: "Cylindrical polar coordinates"
firstLetter: "C"
publishDate: 2021-07-26
categories:
- Mathematics
- Physics
date: 2021-07-26T16:08:46+02:00
draft: false
markup: pandoc
---
# Cylindrical polar coordinates
**Cylindrical polar coordinates** are an extension of polar coordinates to 3D,
which describes the location of a point in space
using the coordinates $(r, \varphi, z)$.
The $z$-axis is unchanged from Cartesian coordinates,
hence it is called a *cylindrical* system.
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:
$$\begin{aligned}
\boxed{
r = \sqrt{x^2 + y^2}
\qquad
\varphi = \mathtt{atan2}(y, x)
\qquad
z = z
}
\end{aligned}$$
The cylindrical polar coordinates form an orthogonal
[curvilinear](/know/concept/curvilinear-coordinates/) system,
whose scale factors $h_r$, $h_\varphi$ and $h_z$ we want to find.
To do so, we calculate the differentials of the Cartesian coordinates:
$$\begin{aligned}
\dd{x} = \dd{r} \cos\varphi - \dd{\varphi} r \sin\varphi
\qquad
\dd{y} = \dd{r} \sin\varphi + \dd{\varphi} r \cos\varphi
\qquad
\dd{z} = \dd{z}
\end{aligned}$$
And then we calculate the line element $\dd{\ell}^2$,
skipping many terms thanks to orthogonality,
$$\begin{aligned}
\dd{\ell}^2
&= \dd{r}^2 \big( \cos^2(\varphi) + \sin^2(\varphi) \big)
+ \dd{\varphi}^2 \big( r^2 \sin^2(\varphi) + r^2 \cos^2(\varphi) \big)
+ \dd{z}^2
\\
&= \dd{r}^2 + r^2 \: \dd{\varphi}^2 + \dd{z}^2
\end{aligned}$$
Finally, we can simply read off
the squares of the desired scale factors
$h_r^2$, $h_\varphi^2$ and $h_z^2$:
$$\begin{aligned}
\boxed{
h_r = 1
\qquad
h_\varphi = r
\qquad
h_z = 1
}
\end{aligned}$$
With these factors, we can easily convert things from the Cartesian system
using the standard formulae for orthogonal curvilinear coordinates.
The basis vectors are:
$$\begin{aligned}
\boxed{
\begin{aligned}
\vu{e}_r
&= \cos\varphi \:\vu{e}_x + \sin\varphi \:\vu{e}_y
\\
\vu{e}_\varphi
&= - \sin\varphi \:\vu{e}_x + \cos\varphi \:\vu{e}_y
\\
\vu{e}_z
&= \vu{e}_z
\end{aligned}
}
\end{aligned}$$
The basic vector operations (gradient, divergence, Laplacian and curl) are given by:
$$\begin{aligned}
\boxed{
\nabla f
= \vu{e}_r \pdv{f}{r}
+ \vu{e}_\varphi \frac{1}{r} \pdv{f}{\varphi}
+ \mathbf{e}_z \pdv{f}{z}
}
\end{aligned}$$
$$\begin{aligned}
\boxed{
\nabla \cdot \vb{V}
= \frac{1}{r} \pdv{(r V_r)}{r}
+ \frac{1}{r} \pdv{V_\varphi}{\varphi}
+ \pdv{V_z}{z}
}
\end{aligned}$$
$$\begin{aligned}
\boxed{
\nabla^2 f
= \frac{1}{r} \pdv{r} \Big( r \pdv{f}{r} \Big)
+ \frac{1}{r^2} \pdv[2]{f}{\varphi}
+ \pdv[2]{f}{z}
}
\end{aligned}$$
$$\begin{aligned}
\boxed{
\begin{aligned}
\nabla \times \vb{V}
&= \vu{e}_r \Big( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \Big)
\\
&+ \vu{e}_\varphi \Big( \pdv{V_r}{z} - \pdv{V_z}{r} \Big)
\\
&+ \frac{\vu{e}_\varphi}{r} \Big( \pdv{(r V_\varphi)}{r} - \pdv{V_r}{\varphi} \Big)
\end{aligned}
}
\end{aligned}$$
The differential element of volume $\dd{V}$
takes the following form:
$$\begin{aligned}
\boxed{
\dd{V}
= r \dd{r} \dd{\varphi} \dd{z}
}
\end{aligned}$$
So, for example, an integral over all of space is converted like so:
$$\begin{aligned}
\iiint_{-\infty}^\infty f(x, y, z) \dd{V}
= \int_{-\infty}^{\infty} \int_0^{2\pi} \int_0^\infty f(r, \varphi, z) \: r \dd{r} \dd{\varphi} \dd{z}
\end{aligned}$$
The isosurface elements are as follows, where $S_r$ is a surface at constant $r$, etc.:
$$\begin{aligned}
\boxed{
\begin{aligned}
\dd{S}_r = r \dd{\varphi} \dd{z}
\qquad
\dd{S}_\varphi = \dd{r} \dd{z}
\qquad
\dd{S}_z = r \dd{r} \dd{\varphi}
\end{aligned}
}
\end{aligned}$$
Similarly, the normal vector element $\dd{\vu{S}}$ for an arbitrary surface is given by:
$$\begin{aligned}
\boxed{
\dd{\vu{S}}
= \vu{e}_r \: r \dd{\varphi} \dd{z}
+ \vu{e}_\varphi \dd{r} \dd{z}
+ \vu{e}_z \: r \dd{r} \dd{\varphi}
}
\end{aligned}$$
And finally, the tangent vector element $\dd{\vu{\ell}}$ of a given curve is as follows:
$$\begin{aligned}
\boxed{
\dd{\vu{\ell}}
= \vu{e}_r \dd{r}
+ \vu{e}_\varphi \: r \dd{\varphi}
+ \vu{e}_z \dd{z}
}
\end{aligned}$$
## References
1. M.L. Boas,
*Mathematical methods in the physical sciences*, 2nd edition,
Wiley.
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