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 . 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 is denoted as θ. Also, the radial coordinate is called r while the vertical coordinate is sometimes referred as h.
A point P is given as . In terms of the Cartesian coordinate system:
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 has the very simple equation in cylindrical coordinates. Hence the name "cylindrical" coordinates.
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
The volume element is
The surface element is
The del operator in this system is written as
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:
where 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:
and Laplace's equation, divided by V, is written:
The Z part of the equation is a function of z alone, and must therefore be equal to a constant:
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:
or by their behavior at infinity:
If k is imaginary:
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 k2 for , Laplace's equation may now be written:
Multiplying by ρ2, we may now separate the P and Φ functions and introduce another constant (n) to obtain:
Since is periodic, we may take n to be a non-negative integer and accordingly, the the constants are subscripted. Real solutions for are
The differential equation for ρ is a form of Bessel's equation.
If k is zero, but n is not, the solutions are:
If both k and n are zero, the solutions are:
If k is a real number we may write a real solution as:
where Jn(z) and Yn(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:
where In(z) and Kn(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:
where the An(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 Jn(x) is often very useful when finding a solution to a particular problem. The and Z(k,z) functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions. When Pn(kρ) is simply Jn(kρ) , the orthogonality of Jn, along with the orthogonality relationships of and Z(k,z) allow the constants to be determined.
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.
As an example, consider the problem of determining the potential of a unit source located at 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 / 4πε0 = 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 Pn(kρ) function to be the ordinary Bessel function Jn(kρ), 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:
where knra is the r-th zero of Jn(z) and, from the orthogonality relationships for each of the functions:
Above the source point:
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 = z0.
Removing the plane ends (i. e. taking the limit as L approaches infinity) gives the field of the point source inside a conducting cylinder:
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:
and R is the distance from the point source to the measurement point:
Finally, when the point source is at the origin, ρ0 = z0 = 0