Cylindrical coordinate system

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Unique cylindrical coordinates[edit]







Coordinate system conversions[edit]

The cylindrical coordinate system is one of many three-dimensional coordinate systems. The following formula may be used to convert between them.

Cartesian coordinates[edit]


x = \rho \cos \varphi
y = \rho \sin \varphi

in one direction, and

\varphi = 
   0 & \mbox{if } x = 0 \mbox{ and } y = 0\\
    \arcsin(\frac{y}{\rho}) & \mbox{if } x \geq 0 \\
    -\arcsin(\frac{y}{\rho}) + \pi & \mbox{if } x < 0\\



Spherical coordinates[edit]


θ is elevation: θ is inclination:
 \rho = r \cos \theta \,  \rho = r \sin \theta \,
 \varphi  = \varphi \,  \varphi  = \varphi \,
 z  = r \sin \theta \,  z  = r \cos \theta \,

Cylindrical coordinates may be converted into spherical coordinates by:

θ is elevation: θ is inclination:
r=\sqrt{\rho^2+z^2} r=\sqrt{\rho^2+z^2}
{\theta}=\operatorname{arcsin}(z/r) {\theta}=\operatorname{arccos}(z/r)
{\varphi}=\varphi \quad {\varphi}=\varphi \quad

Line and volume elements[edit]



The line element is

\mathrm d\mathbf{r} = \mathrm d\rho\,\boldsymbol{\hat \rho} + \rho\,\mathrm d\varphi\,\boldsymbol{\hat\varphi} + \mathrm dz\,\mathbf{\hat z}.

The volume element is

\mathrm{d}V = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z.

The surface element in a surface of constant radius \rho (a vertical cylinder) is

\mathrm{d}S_\rho= \rho\,\mathrm{d}\varphi\,\mathrm{d}z.

The surface element in a surface of constant azimuth \varphi (a vertical half-plane) is

\mathrm{d}S_\varphi= \mathrm{d}\rho\,\mathrm{d}z.


\mathrm{d}S_z= \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi.

The del operator in this system is written as

\nabla = \boldsymbol{\hat \rho}\frac{\partial}{\partial \rho} + \boldsymbol{\hat \varphi}\frac{1}{\rho}\frac{\partial}{\partial \varphi} + \mathbf{\hat z}\frac{\partial}{\partial z},

and the Laplace operator \nabla^2 is defined by

 \nabla^2 f

Cylindrical harmonics[edit]


See also[edit]


  1. ^ Krafft, C.; Volokitin, A. S. (1 January 2002). "Resonant electron beam interaction with several lower hybrid waves". Physics of Plasmas 9 (6): 2786–2797. doi:10.1063/1.1465420. ISSN 1089-7674. Retrieved 9 February 2013. " cylindrical coordinates (r,θ,z) ... and Z=vbzt is the longitudinal position..." 
  2. ^ Alexander Groisman and Victor Steinberg (1997), Solitary Vortex Pairs in Viscoelastic Couette Flow. Physical Review Letters, volume 78, number 8, 1460–1463. doi:10.1103/PhysRevLett.78.1460 "[...]where r, θ, and z are cylindrical coordinates [...] as a function of axial position[...]"

Further reading[edit]

External links[edit]