Definition of Cylindrical Symmetry
Matthew Harrington 
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 $f(\rho,z,\theta)=g(\rho)$.
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$\begingroup$Your definition is my understanding of the term and it appears on all sorts of pages that come up when you Google it, like
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"
$\endgroup$ 8 $\begingroup$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:
$$\{ \rho (\theta) \cos \theta,\; \rho (\theta)\sin\theta, \;f^{-1}\rho(\theta)\}. $$
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