Youtube

Go to The Main Page Add Youtube to favorite!

Paraboloidal coordinates 

Paraboloidal coordinates are a three-dimensional orthogonal coordinate system (λ,μ,ν) 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.

Contents

Basic formulae

The Cartesian coordinates (x,y,z) can be produced from the ellipsoidal coordinates (λ,μ,ν) by the equations

x^{2} = \frac{\left( A - \lambda \right) \left( A - \mu \right) \left( A - \nu \right)}{B - A}
y^{2} = \frac{\left( B - \lambda \right) \left( B - \mu \right) \left( B - \nu \right)}{A - B}
z = \frac{1}{2} \left( A + B - \lambda - \mu -\nu \right)

where the following limits apply to the coordinates

λ < B < μ < A < ν

Consequently, surfaces of constant λ are elliptic paraboloids

\frac{x^{2}}{\lambda - A} + \frac{y^{2}}{\lambda - B} = 2z + \lambda

and surfaces of constant ν are likewise

\frac{x^{2}}{\nu - A} + \frac{y^{2}}{\nu - B} = 2z + \nu

whereas surfaces of constant μ are hyperbolic paraboloids

\frac{x^{2}}{\mu - A} + \frac{y^{2}}{\mu - B} = 2z + \mu

Scale factors

The scale factors for the paraboloidal coordinates (λ,μ,ν) are

h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right)}{ \left( A - \lambda \right) \left( B - \lambda \right)}}
h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \mu \right) \left( \lambda - \mu \right)}{ \left( A - \mu \right) \left( B - \mu \right)}}
h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \nu \right) \left( \mu - \nu \right)}{ \left( A - \nu \right) \left( B - \nu \right)}}

Hence, the infinitesimal volume element equals

dV = \frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right) \left( \nu - \mu\right)}{8\sqrt{\left( A - \lambda \right) \left( B - \lambda \right) \left( A - \mu \right) \left( \mu - B \right) \left( \nu - A \right) \left( \nu - B \right) }} \ d\lambda d\mu d\nu

Differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (λ,μ,ν) by substituting the scale factors into the general formulae found in orthogonal coordinates.

See also

  • Three dimensional orthogonal coordinate systems


References

Bibliography

  • Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand, pp. 184–185. LCCN 55-10911. 
  • Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, p. 180. LCCN 59-14456, 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 67-25285. 
  • 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 978-0387184302. 

External links
Could not update stat
UP