Definition of Cylindrical Symmetry

Question

In cylindrical coordinates I have a function $f(\rho,z,\theta)$. What does it mean for $f$ to be cylindrically symmetric? I can't find any precise definition online. My guess is that $f(\rho,z,\theta)=g(\rho)$.

Answer 1

Your definition is my understanding of the term and it appears on all sorts of pages that come up when you Google it, like

http://www.physicsbootcamp.org/Electric-Field-for-Cylindrical-Symmetry.html

However, it looks like Wikipedia and a number of other sources would call anything of the form $g(\rho,z)$ cylindrically symmetric, saying that cylindrical symmetry merely means "no change when rotating about one axis"

https://en.wikipedia.org/wiki/Rotational_symmetry

Answer 2

Yes, it is rotational symmetry, referring to $(\rho,\theta, z)$ coordinates of the surface and lines drawn on a surface of revolution... the situation is a.k.a axi-symmetry:

A meridian $\rho= f(z)$ (red) is rotated around an axis of symmetry $\rho=0$ along which $z$ is an independent variable. Shown are black circumferential lines.

The meridian is defined

$$\rho= f(z)$$

To locate a point on this surface of revolution we resolve/project $\rho$ further onto $(x,y)$ axes.

The projection enables definition of axi-symmetry for parametrization of any point on a surface of revolution in the cylindrical coordinate system using two parameters:

$$ \{\rho \cos \theta, \rho \sin\theta,f^{-1}(\rho)\} $$

where $ (\rho, \theta)$ can be functions of a fourth parameter say arc length or time for a 3D surface parametric form. This form defines non-meridional spiral situation:

$$ \{\rho (s) \cos \theta(s),\; \rho (s)\sin\theta(s),\; f^{-1}\rho(s)\}. $$

$$\{ \rho (t) \cos \theta(t),\; \rho (t)\sin\theta(t), \;f^{-1}\rho(t)\} $$

This can be also written as a special useful case for as 3D parametric line with a single parameter for the non- meridional situation:

$$\{ \rho (\theta) \cos \theta,\; \rho (\theta)\sin\theta, \;f^{-1}\rho(\theta)\}. $$