A system of Curvilinear Coordinates in which two sets of coordinate surfaces are obtained by revolving the parabolas
of Parabolic Cylindrical Coordinates about the *x*-Axis, which is then relabeled the *z*-Axis. There
are several notational conventions. Whereas
is used in this work,
Arfken (1970) uses
.

The equations for the parabolic coordinates are

(1) | |||

(2) | |||

(3) |

where , , and . To solve for , , and , examine

(4) |

so

(5) |

(6) |

(7) |

(8) | |||

(9) | |||

(10) |

The Scale Factors are

(11) | |||

(12) | |||

(13) |

The Line Element is

(14) |

(15) |

(16) |

The Helmholtz Differential Equation is Separable in parabolic coordinates.

**References**

Arfken, G. ``Parabolic Coordinates (, , ).'' §2.12 in
*Mathematical Methods for Physicists, 2nd ed.* Orlando, FL: Academic Press, pp. 109-112, 1970.

Morse, P. M. and Feshbach, H. *Methods of Theoretical Physics, Part I.* New York:
McGraw-Hill, p. 660, 1953.

© 1996-9

1999-05-26