Bipolar coordinates
The numbers and
which are connected with the Cartesian orthogonal coordinates
and
by the formulas
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where . The coordinate lines are two families of circles
with poles
and
and the (half-c)ircles orthogonal with these
.
Figure: b016470a
The Lamé coefficients are:
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The Laplace operator is:
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Bipolar coordinates in space (bispherical coordinates) are the numbers and
, which are connected with the orthogonal Cartesian coordinates
and
by the formulas:
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where . The coordinate surfaces are spheres (
), the surfaces obtained by rotation of arcs of circles (
) and half-planes passing through the
-axis. The system of bipolar coordinates in space is formed by rotating the system of bipolar coordinates on the plane
around the
-axis.
The Lamé coefficients are:
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The Laplace operator is:
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References
[1] | E. Madelung, "Die mathematischen Hilfsmittel des Physikers" , Springer (1957) |
Comments
References
[a1] | O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1932) |
Bipolar coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bipolar_coordinates&oldid=11655