Cylindrical coordinate system

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 $z$ ) which measures the height of a point above the plane. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by

The notation for this coordinate system is not uniform. The Standard ISO 31-11 establishes them as $(\rho,\varphi,z)$ . ISO 31-11 is the part of International standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. Nevertheless, in many cases the azimuth $\varphi$ is denoted as $θ$ . Also, the radial coordinate is called $r$ while the vertical coordinate is sometimes referred as $h$ .

The coordinate surfaces of the cylindrical coordinates (ρ, φ, z). The coordinate surfaces of a three dimensional Coordinate system are the Surfaces on which a particular coordinate of the system is constant while the coordinate 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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 0, -1. 732, 1. 0).

A point P is given as $(\rho, \varphi, z)$ . In terms of the Cartesian coordinate system:

$ρ$ is the distance from O to P', the orthogonal projection of the point P onto the XY plane. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane This is the same as the distance of P to the z-axis.
$\varphi$ is the angle between the positive x-axis and the line OP', measured counterclockwise.
$z$ is the same as the Cartesian coordinate $z$ .
Thus, the conversion function $f$ from Cartesian coordinates to cylindrical coordinates is $f(\rho,\varphi,z)=(\sqrt{x^{2}+y^{2}},\operatorname{atan2}(y,x),z)\,$ .
The conversion function $f$ from cylindrical coordinates to Cartesian coordinates is $f(x,y,z)=(\rho\cos\varphi,\rho\sin\varphi,z)\,$ .

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 $\ x^2+y^2=c^2$ has the very simple equation $\ \rho = c$ 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 formulae. The multiple integral is a type of definite Integral extended to functions of more than one real Variable, for example This is a list of some Vector calculus formulae of general use in working with various Coordinate systems Table Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner

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

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

The volume element is

$\mathrm dV = \rho\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm dz.$

The surface element is

$\mathrm dS= \rho\,d\varphi\,dz.$

The del operator in this system is written as

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

Cylindrical Harmonics

Cylindrical harmonics are a set of solutions to Laplace's differential equation expressed in cylindrical coordinates. A line element in Mathematics can most generally be thought of as the square of the change in a position vector in an Affine space equated to the square of the change &nablaDel In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually Each harmonic function V_n(k) consists of the product of three functions:

$V_n(k;\rho,\varphi,z)=P_n(k\rho)\Phi_n(\varphi)Z(k,z)\,$

where $(\rho,\varphi,z)$ are the cylindrical coordinates, and n 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. In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems

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:

$V=P(\rho)\,\Phi(\varphi)\,Z(z)$

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

$\frac{\ddot{P}}{P}+\frac{1}{\rho}\,\frac{\dot{P}}{P}+\frac{1}{\rho^2}\,\frac{\ddot{\Phi}}{\Phi}+\frac{\ddot{Z}}{Z}=0$

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

$\frac{\ddot{Z}}{Z}=k^2$

where k is, in general, a complex number. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted For a particular k, the Z(z) function has two linearly independent solutions. If k is real they are:

$Z(k,z)=\cosh(kz)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sinh(kz)\,$

or by their behavior at infinity:

$Z(k,z)=e^{kz}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-kz}\,$

If k is imaginary:

$Z(k,z)=\cos(|k|z)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sin(|k|z)\,$

or:

$Z(k,z)=e^{i|k|z}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-i|k|z}\,$

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. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic

Substituting $k 2$ for $\ddot{Z}/Z$ , Laplace's equation may now be written:

$\frac{\ddot{P}}{P}+\frac{1}{\rho}\,\frac{\dot{P}}{P}+\frac{1}{\rho^2}\frac{\ddot{\Phi}}{\Phi}+k^2=0$

Multiplying by $ρ 2$ , we may now separate the P and Φ functions and introduce another constant (n) to obtain:

$\frac{\ddot{\Phi}}{\Phi} =-n^2$

$\rho^2\frac{\ddot{P}}{P}+\rho\frac{\dot{P}}{P}+k^2\rho^2=n^2$

Since $\varphi$ is periodic, we may take n to be a non-negative integer and accordingly, the $\Phi(\varphi)$ the constants are subscripted. Real solutions for $\Phi(\varphi)$ are

$\Phi_n=\cos(n\varphi)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\sin(n\varphi)$

or, equivalently:

$\Phi_n=e^{in\varphi}\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,e^{-in\varphi}$

The differential equation for $ρ$ is a form of Bessel's equation.

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

$P_n(0,\rho)=\rho^n\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\rho^{-n}\,$

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

$P_n(k,\rho)=\ln\rho\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,1\,$

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

$P_n(k,\rho)=J_n(k\rho)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,Y_n(k\rho)\,$

where $J n (z)$ and $Y n (z)$ are ordinary Bessel functions. In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical If k is an imaginary number, we may write a real solution as:

$P_n(k,\rho)=I_n(|k|\rho)\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,K_n(|k|\rho)\,$

where $I n (z)$ and $K n (z)$ are modified Bessel functions. In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical 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:

$V(\rho,\varphi,z)=\sum_n \int dk\,\, A_n(k) P_n(k,\rho) \Phi_n(\varphi) Z(k,z)\,$

where the $A n (k)$ 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 $J n (x)$ is often very useful when finding a solution to a particular problem. The $\Phi_n(\varphi)$ and $Z (k, z)$ functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions. When $P n (k ρ)$ is simply $J n (k ρ)$ , the orthogonality of $J n$ , along with the orthogonality relationships of $\Phi_n(\varphi)$ and $Z (k, z)$ allow the constants to be determined.

$\int_0^a J_n(k\rho)J_n(k'\rho)\rho\,d\rho = \frac{1}{k}\delta_{kk'}$

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 $(\rho_0,\varphi_0,z_0)$ inside a conducting "cylindrical box" (e. g. an empty tin can) which is bounded above and below by the planes $z = - L$ and $z = L$ and on the sides by the cylinder $ρ = a$ (Smythe, 1968). (In MKS units, we will assume $q / 4πε 0 = 1$ 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 $P n (k ρ)$ function to be the ordinary Bessel function $J n (k ρ)$ , 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:

$V(\rho,\varphi,z)=\sum_{n=0}^\infty \sum_{r=0}^\infty\, A_{nr} J_n(k_{nr}\rho)\cos(n(\varphi-\varphi_0))\sinh(k_{nr}(L+z))\,\,\,\,\,z\le z_0$

where $k n r a$ is the r-th zero of $J n (z)$ and, from the orthogonality relationships for each of the functions:

$A_{nr}=\frac{4(2-\delta_{n0})}{a^2}\,\,\frac{\sinh k_{nr}(L-z_0)}{\sinh 2k_{nr}L}\,\,\frac{J_n(k_{nr}\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}\,$

Above the source point:

$V(\rho,\varphi,z)=\sum_{n=0}^\infty \sum_{r=0}^\infty\, A_{nr} J_n(k_{nr}\rho)\cos(n(\varphi-\varphi_0))\sinh(k_{nr}(L-z))\,\,\,\,\,z\ge z_0$

$A_{nr}=\frac{4(2-\delta_{n0})}{a^2}\,\,\frac{\sinh k_{nr}(L+z_0)}{\sinh 2k_{nr}L}\,\,\frac{J_n(k_{nr}\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}.\,$

It is clear that when $ρ = a$ or $| z | = L$ , 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 $z = z 0$ .

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:

$V(\rho,\varphi,z)=\sum_{n=0}^\infty \sum_{r=0}^\infty\, A_{nr} J_n(k_{nr}\rho)\cos(n(\varphi-\varphi_0))e^{-k_{nr}|z-z_0|}$

$A_{nr}=\frac{2(2-\delta_{n0})}{a^2}\,\,\frac{J_n(k_{nr}\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}.\,$

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:

$V(\rho,\varphi,z)=\frac{1}{R}=\sum_{n=0}^\infty \int_0^\infty dk\, A_n(k) J_n(k\rho)\cos(n(\varphi-\varphi_0))e^{-k|z-z_0|}$

$A_n(k)=(2-\delta_{n0})J_n(k\rho_0)\,$

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

$R=\sqrt{(z-z_0)^2+\rho^2+\rho_0^2-2\rho\rho_0\cos(\varphi-\varphi_0)}.\,$

Point source in open space at origin

Finally, when the point source is at the origin, $ρ 0 = z 0 = 0$

$V(\rho,\varphi,z)=\frac{1}{\sqrt{\rho^2+z^2}}=\int_0^\infty J_0(k\rho)e^{-k|z|}\,dk.$

Bibliography

Morse PM, Feshbach H (1953). This is a list of canonical coordinate transformations. 2-Dimensional Let (x y be the standard Cartesian coordinates, and r and θ the standard Polar Cylindrical coordinate system Vector fields Vectors are defined in Cylindrical coordinates by (ρφz where ρ is the length of the vector projected In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by Parabolic coordinates are a two-dimensional orthogonal Coordinate system in which the Coordinate lines are Confocal Parabolas A three-dimensional two-center bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal Coordinate system. In Mathematics, biangular coordinates are a Coordinate system for the plane where A and B are two fixed points and the position In Mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers C1 and C2 In Mathematics, hyperbolic coordinates are a useful method of locating points in Quadrant I of the Cartesian plane \{(x y \:\ x > 0\ Elliptic coordinates are a two-dimensional orthogonal Coordinate system in which the Coordinate lines are confocal Ellipses and Hyperbolae In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial Parabolic coordinates are a two-dimensional orthogonal Coordinate system in which the Coordinate lines are Confocal Parabolas A three-dimensional Parabolic cylindrical coordinates are a three-dimensional orthogonal Coordinate system that results from projecting the two-dimensional parabolic coordinate Paraboloidal coordinates are a three-dimensional orthogonal Coordinate system (\lambda \mu \nu that generalizes the two-dimensional parabolic Oblate spheroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional elliptic coordinate system Prolate spheroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional elliptic coordinate system Ellipsoidal coordinates are a three-dimensional orthogonal Coordinate system (\lambda \mu \nu that generalizes the two-dimensional elliptic coordinate Elliptic cylindrical coordinates are a three-dimensional orthogonal Coordinate system that results from projecting the two-dimensional elliptic coordinate Toroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional bipolar coordinate system Bispherical coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional bipolar coordinate system Bipolar cylindrical coordinates are a three-dimensional orthogonal Coordinate system that results from projecting the two-dimensional bipolar coordinate Conical coordinates are a three-dimensional orthogonal Coordinate system consisting of concentric spheres (described by their radius r and by two Philip McCord Morse, ( Aug 6, 1903 - Sep 5, 1985) was an American physicist administrator and pioneer of Operations research (OR Herman Feshbach (born in 1917 in New York City &mdash died 22 December 2000 in Cambridge Massachusetts) was an American physicist Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 6576–657. ISBN 0-07-043316-X, LCCN 52-11515. The Library of Congress Control Number or LCCN is a serially based system of numbering cataloging records in the Library of Congress in the United

Margenau H, Murphy GM (1956). Henry Margenau ( 1901 - February 8, 1997) was a German - US Physicist, and Philosopher of science. The Mathematics of Physics and Chemistry. New York: D. van Nostrand, p. 178. LCCN 55-10911. The Library of Congress Control Number or LCCN is a serially based system of numbering cataloging records in the Library of Congress in the United

Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, pp. 174–175. LCCN 59-14456, ASIN B0000CKZX7. The Library of Congress Control Number or LCCN is a serially based system of numbering cataloging records in the Library of Congress in the United

Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag, p. 95. LCCN 67-25285. The Library of Congress Control Number or LCCN is a serially based system of numbering cataloging records in the Library of Congress in the United

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.

External links

MathWorld description of cylindrical coordinates

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