Cylindrical coordinate system

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A point plotted with cylindrical coordinates

The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted Failed to parse (Missing texvc executable; please see math/README to configure.): z ) which measures the height of a point above the plane.

The notation for this coordinate system is not uniform. The Standard ISO 31-11 establishes them as Failed to parse (Missing texvc executable; please see math/README to configure.): (\rho,\varphi,z) . Nevertheless, in many cases the azimuth Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is denoted as Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

. Also, the radial coordinate is called Failed to parse (Missing texvc executable; please see math/README to configure.): r

while the vertical coordinate is sometimes referred as Failed to parse (Missing texvc executable; please see math/README to configure.): h

Image:Cylindrical coordinate surfaces.png

The coordinate surfaces of the cylindrical coordinates (ρ, φ, z). The red cylinder shows the points with ρ=2, the blue 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 coordinates of P are roughly (1.0, -1.732, 1.0).

A point P is given as Failed to parse (Missing texvc executable; please see math/README to configure.): (\rho, \varphi, z) . In terms of the Cartesian coordinate system:

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

is the distance from O to P', the orthogonal projection of the point P onto the XY plane. This is the same as the distance of P to the z-axis.

Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is the angle between the positive x-axis and the line OP', measured counterclockwise.

Failed to parse (Missing texvc executable; please see math/README to configure.): z

is the same as the Cartesian coordinate Failed to parse (Missing texvc executable; please see math/README to configure.): z

Thus, the conversion function Failed to parse (Missing texvc executable; please see math/README to configure.): f

from cylindrical coordinates to Cartesian coordinates is Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y,z)=(\rho\cos\varphi,\rho\sin\varphi,z)\,

The conversion function Failed to parse (Missing texvc executable; please see math/README to configure.): f

from Cartesian coordinates to cylindrical coordinates is Failed to parse (Missing texvc executable; please see math/README to configure.): f(\rho,\varphi,z)=(\sqrt{x^{2}+y^{2}},\operatorname{atan2}(y,x),z)\,

Note that the atan2() function as used above is not standard: It returns a value between 0 and 2π rather than between -π and π as the standard atan2() function does.

Cylindrical coordinates are useful in analyzing surfaces that are symmetrical about an axis, with the z-axis chosen as the axis of symmetry. For example, the infinitely long circular cylinder that has the Cartesian equation Failed to parse (Missing texvc executable; please see math/README to configure.): \ x^2+y^2=c^2

has the very simple equation Failed to parse (Missing texvc executable; please see math/README to configure.): \ \rho = c
in cylindrical coordinates. Hence the name "cylindrical" coordinates.

1 Line and volume elements
2 Cylindrical Harmonics
- 2.1 Example: Point source inside a conducting cylindrical box
3 See also
4 Bibliography
5 External links

Line and volume elements

See multiple integral for details of volume integration in cylindrical coordinates, and Del in cylindrical and spherical coordinates for vector calculus formula.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm dV = \rho\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm dz

The surface element is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm dS= \rho\,d\varphi\,dz

The del operator in this system is written as

Failed to parse (Missing texvc executable; please see math/README to configure.): \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} .

Cylindrical Harmonics

Cylindrical harmonics are a set of solutions to Laplace's differential equation expressed in cylindrical coordinates. Each harmonic function V_n(k) consists of the product of three functions:

Failed to parse (Missing texvc executable; please see math/README to configure.): V_n(k;\rho,\varphi,z)=P_n(k\rho)\Phi_n(\varphi)Z(k,z)\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): (\rho,\varphi,z)

are the cylindrical coordinates, and m and k are constants which distinguish the members of the set from each other. As a result of the superposition principle applied to Laplace's equation, very general solutions to Laplace's equation can be obtained by linear combinations of these functions.

Since all of the surfaces of constant ρ, φ and z are conicoid, Laplace's equation is separable in cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written:

Failed to parse (Missing texvc executable; please see math/README to configure.): V=P(\rho)\,\Phi(\varphi)\,Z(z)

and Laplace's equation, divided by V, is written:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\ddot{P}}{P}+\frac{1}{\rho}\,\frac{\dot{P}}{P}+\frac{1}{\rho^2}\,\frac{\ddot{\Phi}}{\Phi}+\frac{\ddot{Z}}{Z}=0

The Z part of the equation is a function of z alone, and must therefore be equal to a constant:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\ddot{Z}}{Z}=k^2

where k is, in general, a complex number. For a particular var

Failed to parse (Missing texvc executable; please see math/README to configure.): Z(k,z)=\cosh(kz)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sinh(kz)\,

or by their behavior at infinity:

Failed to parse (Missing texvc executable; please see math/README to configure.): Z(k,z)=e^{kz}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-kz}\,

If k is imaginary:

Failed to parse (Missing texvc executable; please see math/README to configure.): Z(k,z)=\cos(|k|z)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sin(|k|z)\,

or:

Failed to parse (Missing texvc executable; please see math/README to configure.): Z(k,z)=e^{|k|z}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-|k|z}\,

It can be seen that the Z(k,z) functions are the kernels of the Fourier transform or Laplace transform of the Z(z) function and so k may be a discrete variable for periodic boundary conditions, or it may be a continuous variable for non-periodic boundary conditions.

Substituting Failed to parse (Missing texvc executable; please see math/README to configure.): k^2

for Failed to parse (Missing texvc executable; please see math/README to configure.): \ddot{Z}/Z

, Laplace's equation may now be written:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\ddot{P}}{P}+\frac{1}{\rho}\,\frac{\dot{P}}{P}+\frac{1}{\rho^2}\frac{\ddot{\Phi}}{\Phi}+k^2=0

Multiplying by Failed to parse (Missing texvc executable; please see math/README to configure.): \rho^2 , we may now separate the P and Φ functions and introduce another constant (n) to obtain:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\ddot{\Phi}}{\Phi} =-n^2

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho^2\frac{\ddot{P}}{P}+\rho\frac{\dot{P}}{P}+k^2\rho^2=n^2

Since Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is periodic, we may take n to be a non-negative integer and accordingly, the Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi(\varphi)
the constants are subscripted.  Real solutions for Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi(\varphi)
are

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_n=\cos(n\varphi)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sin(n\varphi)

or, equivalently:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_n=e^{in\varphi}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-in\varphi}

The differential equation for Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

is a form of Bessel's equation.

If k is zero, but n is not, the solutions are:

Failed to parse (Missing texvc executable; please see math/README to configure.): P_n(0,\rho)=\rho^n\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\rho^{-n}\,

If both k and n are zero, the solutions are:

Failed to parse (Missing texvc executable; please see math/README to configure.): P_n(k,\rho)=\ln\rho\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,1\,

If k is a real number we may write a real solution as:

Failed to parse (Missing texvc executable; please see math/README to configure.): P_n(k,\rho)=J_n(k\rho)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,Y_n(k\rho)\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): J_n(z)

and Failed to parse (Missing texvc executable; please see math/README to configure.): Y_n(z)
are ordinary Bessel functions. If k  is an imaginary number, we may write a real solution as:

Failed to parse (Missing texvc executable; please see math/README to configure.): P_n(k,\rho)=I_n(|k|\rho)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,K_n(|k|\rho)\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): I_n(z)

and Failed to parse (Missing texvc executable; please see math/README to configure.): K_n(z)
are modified Bessel functions. The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(\rho,\varphi,z)=\sum_n \int dk\,\, A_n(k) P_n(k,\rho) \Phi_n(\varphi) Z(k,z)\,

where the Failed to parse (Missing texvc executable; please see math/README to configure.): A_n(k)

are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary conditions of the problem. Note that the integral may be replaced by a sum for appropriate boundary conditions. The orthogonality of the Failed to parse (Missing texvc executable; please see math/README to configure.): J_n(x)
is often very useful when finding a solution to a particular problem. The Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_n(\varphi)
 and Failed to parse (Missing texvc executable; please see math/README to configure.): Z(k,z)
 functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions. When Failed to parse (Missing texvc executable; please see math/README to configure.): P_n(k\rho)
 is simply Failed to parse (Missing texvc executable; please see math/README to configure.): J_n(k\rho)
, the orthogonality of Failed to parse (Missing texvc executable; please see math/README to configure.): J_n

, along with the orthogonality relationships of Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_n(\varphi)

and Failed to parse (Missing texvc executable; please see math/README to configure.): Z(k,z)
allow the constants to be determined.

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^a J_n(k\rho)J_n(k'\rho)\rho\,d\rho = \frac{1}{k}\delta_{kk'}

see smythe p 185 for more orthogonality

In solving problems, the space may be divided into any number of pieces, as long as the values of the potential and its derivative match across a boundary which contains no sources.

Example: Point source inside a conducting cylindrical box

As an example, consider the problem of determining the potential of a unit source located at Failed to parse (Missing texvc executable; please see math/README to configure.): (\rho_0,\varphi_0,z_0)

inside a conducting "cylindrical box" (e.g. an empty tin can) which is bounded above and below by the planes Failed to parse (Missing texvc executable; please see math/README to configure.): z=-L
and Failed to parse (Missing texvc executable; please see math/README to configure.): z=L
and on the sides by the cylinder Failed to parse (Missing texvc executable; please see math/README to configure.): \rho=a
(Smythe, 1968). (In MKS units, we will assume Failed to parse (Missing texvc executable; please see math/README to configure.): q/4\pi\epsilon_0=1
Since the potential is bounded by the planes on the z axis, the Z(k,z) function can be taken to be periodic. Since the potential must be zero at the origin, we take the Failed to parse (Missing texvc executable; please see math/README to configure.): P_n(k\rho)
function to be the ordinary Bessel function Failed to parse (Missing texvc executable; please see math/README to configure.): J_n(k\rho)

, and it must be chosen so that one of its zeroes lands on the bounding cylinder. For the measurement point below the source point on the z axis, the potential will be:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(\rho,\varphi,z)=\sum_{n=0}^\infty \sum_{r=0}^\infty\, A_{nr} J_n(k_{nr}\rho)\cos(n(\varphi-\varphi_0))\sinh(k_{nr}(L+z))\,\,\,\,\,z\le z_0

where Failed to parse (Missing texvc executable; please see math/README to configure.): k_{nr}a

is the r-th zero of Failed to parse (Missing texvc executable; please see math/README to configure.): J_n(z)
and, from the orthogonality relationships for each of the functions:

Failed to parse (Missing texvc executable; please see math/README to configure.): A_{nr}=\frac{4(2-\delta_{n0})}{a^2}\,\,\frac{\sinh k_{nr}(L-z_0)}{\sinh 2k_{nr}L}\,\,\frac{J_n(k_{nr}\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}\,

Above the source point:

Failed to parse (Missing texvc executable; please see math/README to configure.): A_{nr}=\frac{4(2-\delta_{n0})}{a^2}\,\,\frac{\sinh k_{nr}(L+z_0)}{\sinh 2k_{nr}L}\,\,\frac{J_n(k_{nr}\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}\,

It is clear that when Failed to parse (Missing texvc executable; please see math/README to configure.): \rho=a

or Failed to parse (Missing texvc executable; please see math/README to configure.): |z|=L

, the above function is zero. It can also be easily shown that the two functions match in value and in the value of their first derivatives at Failed to parse (Missing texvc executable; please see math/README to configure.): z=z_0 .

Point source inside cylinder

Removing the plane ends (i.e. taking the limit as L approaches infinity) gives the field of the point source inside a conducting cylinder:

Failed to parse (Missing texvc executable; please see math/README to configure.): A_{nr}=\frac{2(2-\delta_{n0})}{a^2}\,\,\frac{J_n(k_{nr}\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}\,

Point source in open space

As the radius of the cylinder (a) approaches infinity, the sum over the zeroes of J_n(z) becomes an integral, and we have the field of a point source in infinite space:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(\rho,\varphi,z) =\frac{1}{R} =\sum_{n=0}^\infty \int_0^\infty dk\, A_n(k) J_n(k\rho)\cos(n(\varphi-\varphi_0))e^{-k|z-z_0|}

Failed to parse (Missing texvc executable; please see math/README to configure.): A_n(k)=(2-\delta_{n0})J_n(k\rho_0)\,

and R is the distance from the point source to the measurement point:

Failed to parse (Missing texvc executable; please see math/README to configure.): R=\sqrt{(z-z_0)^2+\rho^2+\rho_0^2-2\rho\rho_0\cos(\varphi-\varphi_0)}\,

Point source in open space at origin

Finally, when the point source is at the origin, Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_0=z_0=0

Failed to parse (Missing texvc executable; please see math/README to configure.): V(\rho,\varphi,z)=\frac{1}{\sqrt{\rho^2+z^2}}=\int_0^\infty J_0(k\rho)e^{-k|z|}\,dk

Bibliography

Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 6576–657. ISBN 0-07-043316-X, LCCN 52-11515.

Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand, p. 178. LCCN 55-10911.

Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, pp. 174–175. LCCN 59-14456, ASIN B0000CKZX7.

Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag, p. 95. LCCN 67-25285.

Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett, p. 113. ISBN 0-86720-293-9.

Moon P, Spencer DE (1988). "Circular-Cylinder Coordinates (r, ψ, z)", Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, corrected 2nd ed., 3rd print ed., New York: Springer-Verlag, pp. 12–17 (Table 1.02). ISBN 978-0387184302.