A system of curvilinear coordinates. There are several different conventions for the orientation and designation of these coordinates.
Arfken (1970) defines coordinates 
such that
In this work, following Morse and Feshbach (1953), the coordinates 
are used instead. In this convention, the traces of
the coordinate surfaces of the 
-plane are confocal parabolas
with a common axis. The 
curves open into the negative x-axis; the 
curves open into the positive x-axis. The 
and 
curves intersect along the
y-axis.
where 
,
,
and 
.
The scale factors are
 |
(10) |
The Helmholtz differential equation is separable in parabolic cylindrical coordinates.
See also
Confocal Paraboloidal Coordinates, Helmholtz Differential Equation--Parabolic Cylindrical Coordinates, Parabolic Coordinates
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References
Arfken, G. "Parabolic Cylinder Coordinates (
, 
, 
)." ยง2.8 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, p. 97,
1970.Moon, P. and Spencer, D. E. "Parabolic-Cylinder Coordinates
."
Table 1.04 in Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, pp. 21-24, 1988.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, p. 658, 1953.
Referenced on Wolfram|Alpha
Parabolic Cylindrical Coordinates
Cite this as:
Weisstein, Eric W. "Parabolic Cylindrical Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ParabolicCylindricalCoordinates.html