Cylindrical coordinates -Curvilinear

[chwala]

TL;DR Summary

Why are the coordinates seemingly used when the symmetry is around $z$ axis? Any particular reason why not $x$ or $y$. In transforming from Cartesian to cylindrical form; I can see that $z$ is not considered when determining $r$.
Can we also use $x$ and $z$ assuming that the symmetry is about $y$? Sorry using phone to type ...will put this into context later. I hope my query is clear enough.

Why are the coordinates seemingly used when the symmetry is around $z$ axis? Any particular reason why not $x$ or $y$. In transforming from Cartesian to cylindrical form; I can see that $z$ is not considered when determining $r$.
Can we also use $x$ and $z$ assuming that the symmetry is about $y$? Sorry using phone to type ...will put this into context later. I hope my query is clear enough.

 

[pasmith]

The directions of the cartesian axes in space are not absolute, but can be chosen to fit the geometry of the particular problem; by convention in axisymmetric geometries the z-axis is placed on the axis of symmetry, giving an obvious extension of plane polar coordinates $(x,y) = (r \cos \theta, r \sin \theta)$.

It is clearly possible to set $$\begin{split} x &= w \\ y &= u \cos v \\ z &= u \sin v\end{split}$$ or $$\begin{split} x &= u \sin v \\ y &= w \\ z &= u \cos v \end{split}$$ if you want.