Categories: Mathematics, Physics.

Polar cylindrical coordinates

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 $(x, y, z)$ 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, 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.