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Parabolic Coordinates

ParabolicCoordinates

A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the parabolas of parabolic cylindrical coordinates about the x-axis, which is then relabeled the z-axis. There are several notational conventions. Whereas (u,v,theta) is used in this work, Arfken (1970) uses (xi,eta,phi).

The equations for the parabolic coordinates are

x=uvcostheta
(1)
y=uvsintheta
(2)
z=1/2(u^2-v^2),
(3)

where u in [0,infty), v in [0,infty), and theta in [0,2pi). To solve for u, v, and theta, examine

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

so

sqrt(x^2+y^2+z^2)=1/2(u^2+v^2)
(7)

and

sqrt(x^2+y^2+z^2)+z=u^2
(8)
sqrt(x^2+y^2+z^2)-z=v^2.
(9)

We therefore have

u=sqrt(sqrt(x^2+y^2+z^2)+z)
(10)
v=sqrt(sqrt(x^2+y^2+z^2)-z)
(11)
theta=tan^(-1)(y/x).
(12)

The scale factors are

h_u=sqrt(u^2+v^2)
(13)
h_v=sqrt(u^2+v^2)
(14)
h_theta=uv.
(15)

The line element is

ds^2=(u^2+v^2)(du^2+dv^2)+u^2v^2dtheta^2,
(16)

and the volume element is

dV=uv(u^2+v^2)dudvdtheta.
(17)

The Laplacian is

del ^2f=1/(uv(u^2+v^2))[partial/(partialu)(uv(partialf)/(partialu))+partial/(partialv)(uv(partialf)/(partialv))]+1/(u^2v^2)(partial^2f)/(partialtheta^2)
(18)
=1/(u^2+v^2)[1/upartial/(partialu)(u(partialf)/(partialu))+1/vpartial/(partialv)(v(partialf)/(partialv))]+1/(u^2v^2)(partial^2f)/(partialtheta^2)
(19)
=1/(u^2+v^2)(1/u(partialf)/(partialu)+(partial^2f)/(partialu^2)+1/v(partialf)/(partialv)+(partial^2f)/(partialv^2))+1/(u^2v^2)(partial^2f)/(partialtheta^2).
(20)

The Helmholtz differential equation is separable in parabolic coordinates.

See also

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

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References

Arfken, G. "Parabolic Coordinates (xi, eta, phi)." §2.12 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109-112, 1970.Moon, P. and Spencer, D. E. "Parabolic Coordinates (mu,nu,psi)." Table 1.08 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 34-36, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 660, 1953.

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Parabolic Coordinates

Cite this as:

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

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