Paraboloidal coordinates

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Paraboloidal coordinates are a three-dimensional orthogonal coordinate system that generalizes the two-dimensional parabolic coordinate system. Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate surfaces that are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

Coordinate surfaces of the three-dimensional paraboloidal coordinates.

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 elliptic paraboloids

and surfaces of constant are likewise

whereas surfaces of constant are hyperbolic paraboloids

Scale factors[edit]

The scale factors for the paraboloidal coordinates are

Hence, the infinitesimal volume element equals

Differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.



  • Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 664. ISBN 0-07-043316-X. LCCN 52011515. 
  • Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 184–185. LCCN 55010911. 
  • Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 180. LCCN 59014456. ASIN B0000CKZX7. 
  • Arfken G (1970). Mathematical Methods for Physicists (2nd ed.). Orlando, FL: Academic Press. pp. 119–120. 
  • Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. p. 98. LCCN 67025285. 
  • Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9.  Same as Morse & Feshbach (1953), substituting uk for ξk.
  • Moon P, Spencer DE (1988). "Paraboloidal Coordinates (μ, ν, λ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 44–48 (Table 1.11). ISBN 978-0-387-18430-2. 

External links[edit]