A general Metric has a Line Element

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

Orthogonal curvilinear coordinates satisfy the additional constraint that

(12) |

(13) |

(14) |

where the latter is the Jacobian.

Orthogonal curvilinear coordinate systems include Bipolar Cylindrical Coordinates, Bispherical Coordinates, Cartesian Coordinates, Confocal Ellipsoidal Coordinates, Confocal Paraboloidal Coordinates, Conical Coordinates, Cyclidic Coordinates, Cylindrical Coordinates, Ellipsoidal Coordinates, Elliptic Cylindrical Coordinates, Oblate Spheroidal Coordinates, Parabolic Coordinates, Parabolic Cylindrical Coordinates, Paraboloidal Coordinates, Polar Coordinates, Prolate Spheroidal Coordinates, Spherical Coordinates, and Toroidal Coordinates. These are degenerate cases of the Confocal Ellipsoidal Coordinates.

**References**

Arfken, G. ``Curvilinear Coordinates'' and ``Differential Vector Operators.'' §2.1 and 2.2 in
*Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 86-90 and 90-94, 1985.

Gradshteyn, I. S. and Ryzhik, I. M. *Tables of Integrals, Series, and Products, 5th ed.* San Diego, CA:
Academic Press, pp. 1084-1088, 1980.

Morse, P. M. and Feshbach, H. ``Curvilinear Coordinates'' and ``Table of Properties of Curvilinear Coordinates.''
§1.3 in *Methods of Theoretical Physics, Part I.* New York: McGraw-Hill, pp. 21-31 and 115-117, 1953.

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1999-05-25