Ellipsoidal coordinates are a threedimensional orthogonal coordinate system that generalizes the twodimensional elliptic coordinate system. Unlike most threedimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is not produced by rotating or projecting any twodimensional orthogonal coordinate system.
Contents
Basic formulae[edit]
The Cartesian coordinates can be produced from the ellipsoidal coordinates by the equations
where the following limits apply to the coordinates
Consequently, surfaces of constant are ellipsoids
whereas surfaces of constant are hyperboloids of one sheet
because the last term in the lhs is negative, and surfaces of constant are hyperboloids of two sheets
because the last two terms in the lhs are negative.
Scale factors and differential operators[edit]
For brevity in the equations below, we introduce a function
where can represent any of the three variables . Using this function, the scale factors can be written
Hence, the infinitesimal volume element equals
and the Laplacian is defined by
Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.
See also[edit]
 Focaloid (shell given by two coordinate surfaces)
References[edit]
Bibliography[edit]
 Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGrawHill. p. 663.
 Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0867202939.
 Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 101–102. LCCN 67025285.
 Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGrawHill. p. 176. LCCN 59014456.
 Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 178–180. LCCN 55010911.
 Moon PH, Spencer DE (1988). "Ellipsoidal Coordinates (η, θ, λ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 40–44 (Table 1.10). ISBN 0387027327.
Unusual convention[edit]
 Landau LD, Lifshitz EM, Pitaevskii LP (1984). Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) (2nd ed.). New York: Pergamon Press. pp. 19–29. ISBN 9780750626347. Uses (ξ, η, ζ) coordinates that have the units of distance squared.
External links[edit]

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