Cylindrical coordinate system#Line and volume elements

The three coordinates ({{mvar|ρ}}, {{mvar|φ}}, {{mvar|z}}) of a point {{mvar|P}} are defined as:

• The radial distance {{mvar|ρ}} is the Euclidean distance from the {{mvar|z}}-axis to the point {{mvar|P}}.
• The azimuth {{mvar|φ}} is the angle between the reference direction on the chosen plane and the line from the origin to the projection of {{mvar|P}} on the plane.
• The axial coordinate or height {{mvar|z}} is the signed distance from the chosen plane to the point {{mvar|P}}.

As in polar coordinates, the same point with cylindrical coordinates {{math|(ρ, φ, z)}} has infinitely many equivalent coordinates, namely {{math|(ρ, φ ± n×360°, z)}} and {{math|(−ρ, φ ± (2n + 1)×180°, z),}} where {{mvar|n}} is any integer. Moreover, if the radius {{mvar|ρ}} is zero, the azimuth is arbitrary.

In situations where someone wants a unique set of coordinates for each point, one may restrict the radius to be non-negative ({{math|ρ ≥ 0}}) and the azimuth {{mvar|φ}} to lie in a specific interval spanning 360°, such as {{math|[−180°,+180°]}} or {{math|[0,360°]}}.

The notation for cylindrical coordinates is not uniform. The ISO standard 31-11 recommends {{math|(ρ, φ, z)}}, where {{mvar|ρ}} is the radial coordinate, {{mvar|φ}} the azimuth, and {{mvar|z}} the height. However, the radius is also often denoted {{mvar|r}} or {{mvar|s}}, the azimuth by {{mvar|θ}} or {{mvar|t}}, and the third coordinate by {{mvar|h}} or (if the cylindrical axis is considered horizontal) {{mvar|x}}, or any context-specific letter.

File:Cylindrical coordinate surfaces.png of the cylindrical coordinates {{math|(ρ, φ, z)}}. The red cylinder shows the points with {{math|ρ {{=}} 2}}, the blue plane shows the points with {{math|z {{=}} 1}}, and the yellow half-plane shows the points with {{math|φ {{=}} −60°}}. The {{mvar|z}}-axis is vertical and the {{mvar|x}}-axis is highlighted in green. The three surfaces intersect at the point {{mvar|P}} with those coordinates (shown as a black sphere); the Cartesian coordinates of {{mvar|P}} are roughly (1.0, −1.732, 1.0).]]

File:Cylindrical coordinate surfaces.gif

In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured counterclockwise as seen from any point with positive height.

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

For the conversion between cylindrical and Cartesian coordinates, it is convenient to assume that the reference plane of the former is the Cartesian {{mvar|xy}}-plane (with equation {{math|z {{=}} 0}}), and the cylindrical axis is the Cartesian {{mvar|z}}-axis. Then the {{mvar|z}}-coordinate is the same in both systems, and the correspondence between cylindrical {{math|(ρ, φ, z)}} and Cartesian {{math|(x, y, z)}} are the same as for polar coordinates, namely

\begin{align}

x &= \rho \cos \varphi \\

y &= \rho \sin \varphi \\

z &= z

\end{align}

in one direction, and

\varphi &= \begin{cases}

\text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\

\arcsin\left(\frac{y}{\rho}\right) & \text{if } x \geq 0 \\

-\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y \ge 0\\

-\arcsin\left(\frac{y}{\rho}\right) - \pi & \mbox{if } x < 0 \text{ and } y < 0

\end{cases} \end{align}

in the other. The arcsine function is the inverse of the sine function, and is assumed to return an angle in the range {{math|[−{{sfrac|π|2}}, +{{sfrac|π|2}}]}} = {{math|[−90°, +90°]}}. These formulas yield an azimuth {{mvar|φ}} in the range {{math|[−180°, +180°]}}.

By using the arctangent function that returns also an angle in the range {{math|[−{{sfrac|π|2}}, +{{sfrac|π|2}}]}} = {{math|[−90°, +90°]}}, one may also compute $\backslash varphi$ without computing $\backslash rho$ first

$$\backslash begin\{align\}$$

\varphi &= \begin{cases}

\text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\

\frac\pi2\frac y

\arctan\left(\frac{y}{x}\right) & \mbox{if } x > 0 \\

\arctan\left(\frac{y}{x}\right)+\pi & \mbox{if } x < 0 \text{ and } y \ge 0\\

\arctan\left(\frac{y}{x}\right)-\pi & \mbox{if } x < 0 \text{ and } y < 0

\end{cases} \end{align}

For other formulas, see the article Polar coordinate system.

Many modern programming languages provide a function that will compute the correct azimuth {{mvar|φ}}, in the range {{math|(−π, π)}}, given x and y, without the need to perform a case analysis as above. For example, this function is called by {{mono|atan2(y, x)}} in the C programming language, and {{mono|(atan y x)}} in Common Lisp.

Spherical coordinates (radius {{mvar|r}}, elevation or inclination {{mvar|θ}}, azimuth {{mvar|φ}}), may be converted to or from cylindrical coordinates, depending on whether {{mvar|θ}} represents elevation or inclination, by the following:

{{hatnote|1= See multiple integral for details of volume integration in cylindrical coordinates, and Del in cylindrical and spherical coordinates for vector calculus formulae.}}

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.

The line element is

$$\backslash mathrm\{d\}\backslash boldsymbol\{r\}\; =\; \backslash mathrm\{d\}\backslash rho\backslash ,\backslash boldsymbol\{\backslash hat\{\backslash rho\}\}\; +\; \backslash rho\backslash ,\backslash mathrm\{d\}\backslash varphi\backslash ,\backslash boldsymbol\{\backslash hat\{\backslash varphi\}\}\; +\; \backslash mathrm\{d\}z\backslash ,\backslash boldsymbol\{\backslash hat\{z\}\}.$$

The volume element is

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

The surface element in a surface of constant radius {{mvar|ρ}} (a vertical cylinder) is

$$\backslash mathrm\{d\}S\_\backslash rho\; =\; \backslash rho\backslash ,\backslash mathrm\{d\}\backslash varphi\backslash ,\backslash mathrm\{d\}z.$$

The surface element in a surface of constant azimuth {{mvar|φ}} (a vertical half-plane) is

$$\backslash mathrm\{d\}S\_\backslash varphi\; =\; \backslash mathrm\{d\}\backslash rho\backslash ,\backslash mathrm\{d\}z.$$

The surface element in a surface of constant height {{mvar|z}} (a horizontal plane) is

$$\backslash mathrm\{d\}S\_z\; =\; \backslash rho\backslash ,\backslash mathrm\{d\}\backslash rho\backslash ,\backslash mathrm\{d\}\backslash varphi.$$

The del operator in this system leads to the following expressions for gradient, divergence, curl and Laplacian:

$$\backslash begin\{align\}$$

\nabla f &= \frac{\partial f}{\partial \rho}\boldsymbol{\hat{\rho}} + \frac{1}{\rho}\frac{\partial f}{\partial \varphi}\boldsymbol{\hat{\varphi}} + \frac{\partial f}{\partial z}\boldsymbol{\hat{z}} \\[8px]

\nabla \cdot \boldsymbol{A} &= \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho A_\rho\right) + \frac{1}{\rho} \frac{\partial A_\varphi}{\partial \varphi} + \frac{\partial A_z}{\partial z} \\[8px]

\nabla \times \boldsymbol{A} &= \left(\frac{1}{\rho}\frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z}\right)\boldsymbol{\hat{\rho}} +

\left(\frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho}\right)\boldsymbol{\hat{\varphi}} +

\frac{1}{\rho}\left(\frac{\partial}{\partial \rho}\left(\rho A_\varphi\right) - \frac{\partial A_\rho}{\partial \varphi}\right) \boldsymbol{\hat{z}} \\[8px]

\nabla^2 f &= \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho}\right) +

\frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2}

\end{align}

The solutions to the Laplace equation in a system with cylindrical symmetry are called cylindrical harmonics.

In a cylindrical coordinate system, the position of a particle can be written as{{cite book |last1=Taylor |first1=John R. |title=Classical Mechanics |date=2005 |publisher=University Science Books |location=Sausalito, California |page=29}}

$$\backslash boldsymbol\{r\}\; =\; \backslash rho\backslash ,\backslash boldsymbol\{\backslash hat\; \backslash rho\}\; +\; z\backslash ,\backslash boldsymbol\{\backslash hat\; z\}.$$

The velocity of the particle is the time derivative of its position,

$$\backslash boldsymbol\{v\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash boldsymbol\{r\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash dot\{\backslash rho\}\backslash ,\backslash boldsymbol\{\backslash hat\; \backslash rho\}\; +\; \backslash rho\backslash ,\backslash dot\backslash varphi\backslash ,\backslash hat\{\backslash boldsymbol\{\backslash varphi\}\}\; +\; \backslash dot\{z\}\backslash ,\backslash hat\{\backslash boldsymbol\{z\}\},$$

where the term $\backslash rho\; \backslash dot\backslash varphi\backslash hat\backslash varphi$ comes from the Poisson formula $\backslash frac\{\backslash mathrm\; d\backslash hat\backslash rho\}\{\backslash mathrm\; dt\}\; =\; \backslash dot\backslash varphi\backslash hat\; z\backslash times\; \backslash hat\backslash rho$. Its acceleration is

\boldsymbol{a} = \frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t} = \left( \ddot{\rho} - \rho\,\dot\varphi^2 \right)\boldsymbol{\hat \rho} + \left( 2\dot{\rho}\,\dot\varphi + \rho\,\ddot\varphi \right) \hat{\boldsymbol\varphi } + \ddot{z}\,\hat{\boldsymbol{z}}

• List of canonical coordinate transformations
• Vector fields in cylindrical and spherical coordinates
• Del in cylindrical and spherical coordinates

{{reflist}}

• {{cite book |last1=Morse |first1=Philip M. |author-link1=Philip M. Morse |last2=Feshbach |first2=Herman |author-link2=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}}
• {{cite book |last1=Margenau |first1=Henry |author-link1=Henry Margenau |last2=Murphy |first2=George M. |year=1956 |title=The Mathematics of Physics and Chemistry |url=https://archive.org/details/mathematicsofphy0002marg |url-access=registration |publisher=D. van Nostrand |location=New York City |page=[https://archive.org/details/mathematicsofphy0002marg/page/178 178] |lccn=55010911 |isbn=9780882754239 |oclc=3017486}}
• {{cite book |last1=Korn |first1=Granino A. |last2=Korn |first2=Theresa M. |author2-link= Theresa M. Korn |year=1961 |title=Mathematical Handbook for Scientists and Engineers |url=https://archive.org/details/mathematicalhand0000korn |url-access=registration |publisher=McGraw-Hill |location=New York City |id=ASIN B0000CKZX7 | pages=[https://archive.org/details/mathematicalhand0000korn/page/174 174–175] | lccn=59014456}}
• {{cite book |last1=Sauer |first1=Robert |last2=Szabó |first2=István |year=1967 |title=Mathematische Hilfsmittel des Ingenieurs |publisher=Springer-Verlag |location=New York City |page=95 |lccn=67025285}}
• {{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}}
• {{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}}

• {{springer|title=Cylinder coordinates|id=p/c027600}}
• [http://mathworld.wolfram.com/CylindricalCoordinates.html MathWorld description of cylindrical coordinates]
• [https://web.archive.org/web/20100708120521/http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm Cylindrical Coordinates] Animations illustrating cylindrical coordinates by Frank Wattenberg

{{Orthogonal coordinate systems}}

Category:Three-dimensional coordinate systems

Category:Orthogonal coordinate systems

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