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| [[Image:Coord system CY 1.svg|thumb|240px|A cylindrical coordinate system with origin ''O'', polar axis ''A'', and longitudinal axis ''L''. The dot is the point with radial distance ''ρ'' = 4, angular coordinate ''φ'' = 130°, and height ''z'' = 4.]]
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| A '''cylindrical coordinate system''' is a three-dimensional [[coordinate system]]
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| that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.
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| The ''origin'' of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis.
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| The axis is variously called the ''cylindrical'' or ''longitudinal'' axis, to differentiate it from
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| the ''polar axis'', which is the [[Line (mathematics)#Ray|ray]] that lies in the reference plane,
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| starting at the origin and pointing in the reference direction.
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| The distance from the axis may be called the ''radial distance'' or ''radius'',
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| while the angular coordinate is sometimes referred to as the ''angular position'' or as the ''azimuth''.
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| The radius and the azimuth are together called the ''polar coordinates'', as they correspond to a two-dimensional [[polar coordinates|polar coordinate]] system in the plane through the point, parallel to the reference plane.
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| The third coordinate may be called the ''height'' or ''altitude'' (if the reference plane is considered horizontal),
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| ''longitudinal position'',<ref>{{cite journal |last1=Krafft |first1=C. |last2=Volokitin |first2=A. S. |title=Resonant electron beam interaction with several lower hybrid waves |journal=Physics of Plasmas |date=1 January 2002 |volume=9 |issue=6 |pages=2786–2797 |doi=10.1063/1.1465420 |url=http://pop.aip.org/resource/1/phpaen/v9/i6/p2786_s1?isAuthorized=no |accessdate=9 February 2013 |issn=1089-7674 |quote=...in cylindrical coordinates (''r'',''θ'',''z'') ... and Z=v<sub>bz</sub>t is the longitudinal position...}}</ref>
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| or ''axial position''.<ref>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}} "<nowiki>[...]</nowiki>where ''r'', ''θ'', and ''z'' are cylindrical coordinates <nowiki>[...]</nowiki> as a function of axial position<nowiki>[...]</nowiki>"</ref>
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| Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational [[symmetry]] about the longitudinal axis, such as water flow in a straight pipe with round cross-section, heat distribution in a metal [[cylinder (geometry)|cylinder]], [[electromagnetic fields]] produced by an [[electric current]] in a long, straight wire, and so on.
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| It is sometimes called "cylindrical polar coordinate"<ref>J. E. Szymanski, ''Basic mathematics for electronic engineers: models and applications'', Volume 16 of Tutorial guides in electronic engineering, Publisher Taylor & Francis, 1989, ISBN 0-278-00068-1, ISBN 978-0-278-00068-1 ([http://books.google.co.uk/books?id=L7wOAAAAQAAJ&lpg=PA170&dq=%22Cylindrical%20polar%20coordinate%22&pg=PA170#v=onepage&q=%22Cylindrical%20polar%20coordinate%22&f=false page 170])</ref> and "polar cylindrical coordinate",<ref>Robert H. Nunn, ''Intermediate fluid mechanics'', Publisher Taylor & Francis, 1989, ISBN 0-89116-647-5, ISBN 978-0-89116-647-4, 343 pages ([http://books.google.co.uk/books?id=0KfkkbX-NYQC&lpg=PA3&dq=%22polar%20Cylindrical%20%20coordinate%22&pg=PA3#v=onepage&q=%22polar%20Cylindrical%20%20coordinate%22&f=false page 3])</ref> and is sometimes used to specify the position of stars in a galaxy ("galactocentric cylindrical polar coordinate").<ref>Linda Siobhan Sparke, John Sill Gallagher, ''Galaxies in the universe: an introduction'', Edition 2, Publisher Cambridge University Press, 2007, ISBN 0-521-85593-4, ISBN 978-0-521-85593-8, 431 pages ([http://books.google.co.uk/books?id=N8Hngab5liQC&lpg=PA37&dq=cylindrical%20polar%20coordinate%20galaxy&pg=PA37#v=onepage&q=cylindrical%20polar%20coordinate%20galaxy&f=false page 37])</ref>
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| ==Definition==
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| The three coordinates (''[[Rho (letter)|ρ]]'', ''[[Phi|φ]]'', ''z'') of a point ''P'' are defined as:
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| * The radial distance ''ρ'' is the [[Euclidean distance]] from the z axis to the point ''P''.
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| * The azimuth ''φ'' is the angle between the reference direction on the chosen plane and the line from the origin to the projection of ''P'' on the plane.
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| * The height ''z'' is the signed distance from the chosen plane to the point P.
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| ===Unique cylindrical coordinates===
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| As in polar coordinates, the same point with cylindrical coordinates (ρ, φ, ''z'') has infinitely many equivalent coordinates, namely {{nowrap|(''ρ'', ''φ'' ± ''n''×360°, ''z'')}} and {{nowrap|(−''ρ'', ''φ'' ± (2''n'' + 1)×180°, ''z''),}} where ''n'' is any integer. Moreover, if the radius ρ is zero, the azimuth is arbitrary.
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| In situations where one needs a unique set of coordinates for each point, one may restrict the radius to be [[non-negative]] (''ρ'' ≥ 0) and the azimuth ''φ'' to lie in a specific [[interval (mathematics)|interval]] spanning 360°, such as (−180°,+180°] or [0,360°).
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| ===Conventions===
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| The notation for cylindrical coordinates is not uniform. The [[International Organization for Standardization|ISO]] standard [[ISO 31-11|31-11]] recommends (''ρ'', ''φ'', ''z''), where ''ρ'' is the radial coordinate, φ the azimuth, and ''z'' the height. However, the radius is also often denoted ''r'', the azimuth by θ or ''t'', and the third coordinate by ''h'' or (if the cylindrical axis is considered horizontal) ''x'', or any context-specific letter.
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| [[File:Cylindrical coordinate surfaces.png|thumb|240px|right|The [[Coordinate system#Coordinate surface|coordinate surfaces]] of the cylindrical coordinates (''ρ'', ''φ'', ''z''). The red [[Cylinder (geometry)|cylinder]] shows the points with ''ρ''=2, the blue [[plane (mathematics)|plane]] shows the points with ''z''=1, and the yellow half-plane shows the points with ''φ''=−60°. The ''z''-axis is vertical and the ''x''-axis is highlighted in green. The three surfaces intersect at the point '''P''' with those coordinates (shown as a black sphere); the [[Cartesian coordinate system|Cartesian coordinates]] of '''P''' are roughly (1.0, −1.732, 1.0).]]
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| [[File:Cylindrical coordinate surfaces.gif|right|thumb|480px|Cylindrical Coordinate Surfaces. The three orthogonal components, ''ρ'' (green), ''φ'' (red), and ''z'' (blue), each increasing at a constant rate. The point is at the intersection between the three colored surfaces.]]
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| In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured [[clockwise|counterclockwise]] as seen from any point with positive height.
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| ==Coordinate system conversions==
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| The cylindrical coordinate system is one of many three-dimensional coordinate systems. The following formula may be used to convert between them.
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| ===Cartesian coordinates===
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| For the conversion between cylindrical and Cartesian coordinate co-ordinates, it is convenient to assume that the reference plane of the former is the Cartesian ''x''–''y'' plane (with equation ''z'' = 0), and the cylindrical axis is the Cartesian ''z'' axis. Then the ''z'' coordinate is the same in both systems, and the correspondence between cylindrical (''ρ'',''φ'') and Cartesian (''x'',''y'') are the same as for polar coordinates, namely
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| :<math>x = \rho \cos \varphi</math>
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| :<math>y = \rho \sin \varphi</math>
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| in one direction, and
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| :<math>\rho = \sqrt{x^{2}+y^{2}}</math>
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| :<math>\varphi =
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| \begin{cases}
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| 0 & \mbox{if } x = 0 \mbox{ and } y = 0\\
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| \arcsin(\frac{y}{\rho}) & \mbox{if } x \geq 0 \\
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| -\arcsin(\frac{y}{\rho}) + \pi & \mbox{if } x < 0\\
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| \end{cases}
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| </math>
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| in the other. The arcsin function is the inverse of the [[trigonometric function|sine]] function, and is assumed to return an angle in the range [−π/2,+π/2] = [−90°,+90°]. These formulas yield an azimuth ''φ'' in the range [−90°,+270°]. For other formulas, see [[Polar coordinate system|the polar coordinate article]].
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| Many modern programming languages provide a function that will compute the correct azimuth ''φ'', in the range (−π, π], given ''x'' and ''y'', without the need to perform a case analysis as above. For example, this function is called by [[atan2|<code>atan2</code>]](''y'',''x'') in the [[C (programming language)|C]] programming language, and [[atan2|<code>atan</code>]](''y'',''x'') in [[Common Lisp]].
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| ===Spherical coordinates===
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| [[Spherical coordinates]] (radius ''r'', elevation or inclination ''θ'', azimuth ''φ''), may be converted into cylindrical coordinates by:
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| {|
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| |-
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| | style="padding-right:8em" | ''θ is elevation:''
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| | ''θ is inclination:''
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| |-
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| | <math> \rho = r \cos \theta \,</math>
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| | <math> \rho = r \sin \theta \,</math>
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| |-
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| | <math> \varphi = \varphi \,</math>
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| | <math> \varphi = \varphi \,</math>
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| |-
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| | <math> z = r \sin \theta \,</math>
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| | <math> z = r \cos \theta \,</math>
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| |}
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| Cylindrical coordinates may be converted into spherical coordinates by:
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| {|
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| |-
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| | style="padding-right:8em" | ''θ is elevation:''
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| | ''θ is inclination:''
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| |-
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| | <math>r=\sqrt{\rho^2+z^2}</math>
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| | <math>r=\sqrt{\rho^2+z^2}</math>
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| |-
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| | <math>{\theta}=\operatorname{arcsin}(z/r)</math>
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| | <math>{\theta}=\operatorname{arccos}(z/r)</math>
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| |-
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| | <math>{\varphi}=\varphi \quad</math>
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| | <math>{\varphi}=\varphi \quad</math>
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| |}
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| ==Line and volume elements==
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| :''See [[Multiple integral#Cylindrical coordinates|multiple integral]] for details of volume integration in cylindrical coordinates, and [[Del in cylindrical and spherical coordinates]] for [[vector calculus]] formulae.''
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| In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.
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| The [[line element]] is
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| :<math>\mathrm d\mathbf{r} = \mathrm d\rho\,\boldsymbol{\hat \rho} + \rho\,\mathrm d\varphi\,\boldsymbol{\hat\varphi} + \mathrm dz\,\mathbf{\hat z}.</math>
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| The [[volume element]] is
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| :<math>\mathrm{d}V = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z.</math>
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| The [[surface element]] in a surface of constant radius <math>\rho</math> (a vertical cylinder) is
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| :<math>\mathrm{d}S_\rho= \rho\,\mathrm{d}\varphi\,\mathrm{d}z.</math>
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| The surface element in a surface of constant azimuth <math>\varphi</math> (a vertical half-plane) is
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| :<math>\mathrm{d}S_\varphi= \mathrm{d}\rho\,\mathrm{d}z.</math>
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| The surface element in a surface of constant height <math>z</math> (a horizontal plane) is
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| :<math>\mathrm{d}S_z= \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi.</math>
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| The [[del]] operator in this system is written as
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| :<math>\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},</math>
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| and the [[Laplace operator]] <math>\nabla^2</math> is defined by
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| :<math> \nabla^2 f
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| = {1 \over \rho} {\partial \over \partial \rho}
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| \left( \rho {\partial f \over \partial \rho} \right)
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| + {1 \over \rho^2} {\partial^2 f \over \partial \varphi^2}
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| + {\partial^2 f \over \partial z^2 }.
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| </math>
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| ==Cylindrical harmonics==
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| The solutions to the [[Laplace equation]] in a system with cylindrical symmetry are called [[cylindrical harmonics]].
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| ==See also==
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| *[[List of canonical coordinate transformations]]
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| *[[Vector fields in cylindrical and spherical coordinates]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{cite book |last=Morse |first=Philip M. |authorlink1=Philip M. Morse |last2=Feshbach |first2=Herman |authorlink2=Herman Feshbach |year=1953 |title=Methods of Theoretical Physics, Part I |publisher=[[McGraw-Hill]] |location=[[New York City]] |isbn=0-07-043316-X |lccn=52011515 |pages=656–657}}
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| *{{cite book |last1=Margenau |first1=Henry |authorlink1=Henry Margenau |last2=Murphy |first2=George M. |year=1956 |title=The Mathematics of Physics and Chemistry |publisher=D. van Nostrand |location=New York City |page=178 |lccn=55010911 |isbn=9780882754239 |oclc=3017486}}
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| *{{cite book |last1=Korn |first1=Granino A. |last2=Korn |first2=Theresa M. |year=1961 |title=Mathematical Handbook for Scientists and Engineers |publisher=McGraw-Hill |location=New York City |id=ASIN B0000CKZX7 | pages=174–175 | lccn=59014456}}
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| *{{cite book |last1=Sauer |first1=Robert |last2=Szabó |first2=István |year=1967 |title=Mathematische Hilfsmittel des Ingenieurs |publisher=[[Springer Science+Business Media|Springer-Verlag]] |location=New York City |page=95 |lccn=67025285}}
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| *{{cite book |last=Zwillinger |first=Daniel |year=1992 |title=Handbook of Integration |publisher=[[Jones and Bartlett Publishers]] |location=[[Boston]] |isbn=0-86720-293-9 |page=113 |oclc=25710023}}
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| *{{cite book |last1=Moon |first1=P. |last2=Spencer |first2=D. E. |year=1988 |chapter=Circular-Cylinder Coordinates (r, ψ, z) |title=Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions |edition=corrected 2nd |publisher=Springer-Verlag |location=New York City |pages=12–17, Table 1.02 |isbn=978-0-387-18430-2}}
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| ==External links==
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| * {{springer|title=Cylinder coordinates|id=p/c027600}}
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| *[http://mathworld.wolfram.com/CylindricalCoordinates.html MathWorld description of cylindrical coordinates]
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| *[http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm Cylindrical Coordinates] Animations illustrating cylindrical coordinates by Frank Wattenberg
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| {{Orthogonal coordinate systems}}
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| [[Category:Coordinate systems]]
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| [[de:Polarkoordinaten#Zylinderkoordinaten]]
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| [[ro:Coordonate polare#Coordonate cilindrice]]
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| [[fi:Koordinaatisto#Sylinterikoordinaatisto]]
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