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Parabolic Cylindrical Coordinates


ParabolicCylindricalCoords

A system of curvilinear coordinates. There are several different conventions for the orientation and designation of these coordinates. Arfken (1970) defines coordinates (xi,eta,z) such that

x=xieta
(1)
y=1/2(eta^2-xi^2)
(2)
z=z.
(3)

In this work, following Morse and Feshbach (1953), the coordinates (u,v,z) are used instead. In this convention, the traces of the coordinate surfaces of the xy-plane are confocal parabolas with a common axis. The u curves open into the negative x-axis; the v curves open into the positive x-axis. The u and v curves intersect along the y-axis.

x=1/2(u^2-v^2)
(4)
y=uv
(5)
z=z,
(6)

where u in [0,infty), v in [0,infty), and z in (-infty,infty). The scale factors are

h_1=sqrt(u^2+v^2)
(7)
h_2=sqrt(u^2+v^2)
(8)
h_3=1.
(9)

Laplace's equation is

 del ^2f=1/(u^2+v^2)((partial^2f)/(partialu^2)+(partial^2f)/(partialv^2))+(partial^2f)/(partialz^2).
(10)

The Helmholtz differential equation is separable in parabolic cylindrical coordinates.


See also

Confocal Paraboloidal Coordinates, Helmholtz Differential Equation--Parabolic Cylindrical Coordinates, Parabolic Coordinates

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References

Arfken, G. "Parabolic Cylinder Coordinates (xi, eta, z)." §2.8 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, p. 97, 1970.Moon, P. and Spencer, D. E. "Parabolic-Cylinder Coordinates (mu,nu,z)." Table 1.04 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 21-24, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 658, 1953.

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Parabolic Cylindrical Coordinates

Cite this as:

Weisstein, Eric W. "Parabolic Cylindrical Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParabolicCylindricalCoordinates.html

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