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Bipolar Cylindrical Coordinates

BipolarCoordinates

A set of curvilinear coordinates defined by

x=(asinhv)/(coshv-cosu)
(1)
y=(asinu)/(coshv-cosu)
(2)
z=z,
(3)

where u in [0,2pi), v in (-infty,infty), and z in (-infty,infty). There are several notational conventions, and whereas (u,v,z) is used in this work, Arfken (1970) prefers (eta,xi,z). The following identities show that curves of constant u and v are circles in xy-space.

x^2+(y-acotu)^2=a^2csc^2u
(4)
(x-acothv)^2+y^2=a^2csch^2v.
(5)

The scale factors are

h_u=a/(coshv-cosu)
(6)
h_v=a/(coshv-cosu)
(7)
h_z=1.
(8)

The Laplacian is

del ^2=((coshv-cosu)^2)/(a^2)((partial^2)/(partialu^2)+(partial^2)/(partialv^2))+(partial^2)/(partialz^2).
(9)

Laplace's equation is not separable in bipolar cylindrical coordinates, but it is in two-dimensional bipolar coordinates.

See also

Bipolar Coordinates, Polar Coordinates

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References

Arfken, G. "Bipolar Coordinates (xi, eta, z)." §2.9 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 97-102, 1970.

Referenced on Wolfram|Alpha

Bipolar Cylindrical Coordinates

Cite this as:

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

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