A system of curvilinear coordinates. There are several different conventions for the orientation and designation of these coordinates. Arfken (1970) defines coordinates such that

			(1)
			(2)
			(3)

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.

			(4)
			(5)
			(6)

where , , and . The scale factors are

			(7)
			(8)
			(9)

Laplace's equation is

(10)

The Helmholtz differential equation is separable in parabolic cylindrical coordinates.

Arfken, G. "Parabolic Cylinder Coordinates (, , )." §2.8 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.

CITE THIS AS:

Weisstein, Eric W. "Parabolic Cylindrical Coordinates." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParabolicCylindricalCoordinates.html