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 [itex](x,y) = (r \cos \theta, r \sin \theta)[/itex].

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