Paraboloidal coordinates

Paraboloidal coordinates are a three-dimensional orthogonal coordinate system $(\lambda ,\mu ,\nu )$ 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

The Cartesian coordinates $(x,y,z)$ can be produced from the ellipsoidal coordinates $(\lambda ,\mu ,\nu )$ 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

$\lambda <B<\mu <A<\nu$

Consequently, surfaces of constant $\lambda$ are elliptic paraboloids

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

and surfaces of constant $\nu$ are likewise

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

whereas surfaces of constant $\mu$ are hyperbolic paraboloids

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

Scale factors

The scale factors for the paraboloidal coordinates $(\lambda ,\mu ,\nu )$ 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 $(\lambda ,\mu ,\nu )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Bibliography

• 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 u_k for ξ_k.
• 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

• MathWorld description of confocal paraboloidal coordinates