“Paraboloidal coordinates”的意思、由来-开放百科全书

词条

Paraboloidal coordinates

释义

Basic formulae
Scale factors
References
Bibliography
External links

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.

Basic formulae

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

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.

References

Bibliography

{{cite book | vauthors = Morse PM, Feshbach H | author-link = Philip M. Morse | author2-link = Herman Feshbach | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | isbn = 0-07-043316-X|lccn=52011515 | page = 664}}
{{cite book | vauthors = Margenau H, Murphy GM | author-link = Henry Margenau | year = 1956 | title = The Mathematics of Physics and Chemistry | publisher = D. van Nostrand | location = New York | pages = 184–185 | lccn = 55010911 }}
{{cite book | vauthors = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | page = 180 | lccn = 59014456}}
{{cite book | author = Arfken G | year = 1970 | title = Mathematical Methods for Physicists | edition = 2nd | publisher = Academic Press | location = Orlando, FL | pages = 119–120}}
{{cite book | vauthors = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | page = 98 | lccn = 67025285}}
{{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | page = 114}} Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
{{cite book | vauthors = Moon P, Spencer DE | year = 1988 | chapter = Paraboloidal Coordinates (μ, ν, λ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | pages = 44–48 (Table 1.11) | isbn = 978-0-387-18430-2}}

External links

MathWorld description of confocal paraboloidal coordinates

1 : Coordinate systems

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