Categories: Mathematics, Physics.

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 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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