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Coordinate surfaces of elliptic cylindrical coordinates. The yellow sheet is the prism of a half-hyperbola corresponding to ν=-45°, whereas the red tube is an elliptical prism corresponding to μ=1. The blue sheet corresponds to z=1. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (2.182, -1.661, 1.0). The foci of the ellipse and hyperbola lie at x = ±2.0.
Coordinate surfaces of elliptic cylindrical coordinates. 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 yellow sheet is the prism of a half-hyperbola corresponding to ν=-45°, whereas the red tube is an elliptical prism corresponding to μ=1. The blue sheet corresponds to z=1. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (2. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 182, -1. 661, 1. 0). The foci of the ellipse and hyperbola lie at x = ±2. 0.

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z-direction. In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Elliptic coordinates are a two-dimensional orthogonal Coordinate system in which the Coordinate lines are confocal Ellipses and Hyperbolae Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. 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 General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions The two foci F1 and F2 are generally taken to be fixed at − a and + a, respectively, on the x-axis of the Cartesian coordinate system. In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

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Basic definition

The most common definition of elliptic cylindrical coordinates (μ,ν,z) is

x = a \ \cosh \mu \ \cos \nu
y = a \ \sinh \mu \ \sin \nu
z = z\!

where μ is a nonnegative real number and \nu \in [0, 2\pi).

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

\frac{x^{2}}{a^{2} \cosh^{2} \mu} + \frac{y^{2}}{a^{2} \sinh^{2} \mu} = \cos^{2} \nu + \sin^{2} \nu = 1

shows that curves of constant μ form ellipses, whereas the hyperbolic trigonometric identity

\frac{x^{2}}{a^{2} \cos^{2} \nu} - \frac{y^{2}}{a^{2} \sin^{2} \nu} = \cosh^{2} \mu - \sinh^{2} \mu = 1

shows that curves of constant ν form hyperbolae. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions

Scale factors

The scale factors for the elliptic cylindrical coordinates μ and ν are equal

h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu}

whereas the remaining scale factor hz = 1. Consequently, an infinitesimal volume element equals

dV = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu dz

and the Laplacian equals

\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right) + \frac{\partial^{2} \Phi}{\partial z^{2}}

Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (μ,ν,z) by substituting the scale factors into the general formulae found in orthogonal coordinates. In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2.

Alternative definition

An alternative and geometrically intuitive set of elliptic coordinates (σ,τ,z) are sometimes used, where σ = coshμ and τ = cosν. Hence, the curves of constant σ are ellipses, whereas the curves of constant τ are hyperbolae. The coordinate τ must belong to the interval [-1, 1], whereas the σ coordinate must be greater than or equal to one.

The coordinates (σ,τ,z) have a simple relation to the distances to the foci F1 and F2. For any point in the (x,y) plane, the sum d1 + d2 of its distances to the foci equals 2aσ, whereas their difference d1 − d2 equals 2aτ. Thus, the distance to F1 is a(σ + τ), whereas the distance to F2 is a(σ − τ). (Recall that F1 and F2 are located at x = − a and x = + a, respectively. )

A drawback of these coordinates is that they do not have a 1-to-1 transformation to the Cartesian coordinates

x = a\sigma\tau \!
y^{2} = a^{2} \left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)

Alternative scale factors

The scale factors for the alternative elliptic coordinates (σ,τ,z) are

h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}}
h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}}

and, of course, hz = 1. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane Hence, the infinitesimal volume element becomes

dV = a^{2} \frac{\sigma^{2} - \tau^{2}}{\sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}} d\sigma d\tau dz

and the Laplacian equals

\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sigma^{2} - \tau^{2} \right) }\left[\sqrt{\sigma^{2} - 1} \frac{\partial}{\partial \sigma} \left( \sqrt{\sigma^{2} - 1} \frac{\partial \Phi}{\partial \sigma} \right) + \sqrt{1 - \tau^{2}} \frac{\partial}{\partial \tau} \left( \sqrt{1 - \tau^{2}} \frac{\partial \Phi}{\partial \tau} \right)\right] + \frac{\partial^{2} \Phi}{\partial z^{2}}

Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (σ,τ) by substituting the scale factors into the general formulae found in orthogonal coordinates. In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2.

Applications

The classic applications of elliptic cylindrical coordinates are in solving partial differential equations, e. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i g. , Laplace's equation or the Helmholtz equation, for which elliptic cylindrical coordinates allow a separation of variables. In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties The Helmholtz equation, named for Hermann von Helmholtz, is the Elliptic partial differential equation (\nabla^2 + k^2 A = 0 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 A typical example would be the electric field surrounding a flat conducting plate of width 2a. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can

The three-dimensional wave equation, when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equations. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves In Mathematics, the Mathieu functions are certain Special functions useful for treating a variety of interesting problems in applied mathematics including

The geometric properties of elliptic coordinates can also be useful. A typical example might involve an integration over all pairs of vectors \mathbf{p} and \mathbf{q} that sum to a fixed vector \mathbf{r} = \mathbf{p} + \mathbf{q}, where the integrand was a function of the vector lengths \left| \mathbf{p} \right| and \left| \mathbf{q} \right|. (In such a case, one would position \mathbf{r} between the two foci and aligned with the x-axis, i. e. , \mathbf{r} = 2a \mathbf{\hat{x}}. ) For concreteness, \mathbf{r}, \mathbf{p} and \mathbf{q} could represent the momenta of a particle and its decomposition products, respectively, and the integrand might involve the kinetic energies of the products (which are proportional to the squared lengths of the momenta). In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product

See also


Bibliography

External links

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