Variously denoted
or .

(1) |

where is a Connection Coefficient and is a Christoffel Symbol of the First Kind.

(2) |

(3) | |||

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) |

and and . If , the Christoffel symbols of the second kind simplify to

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) |

(Gray 1993).

The following relationships hold between the Christoffel symbols of the second kind and coefficients of the first
Fundamental Form,

(15) | |||

(16) | |||

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) | |||

(22) |

(Gray 1993).

For a surface given in Monge's Form ,

(23) |

**References**

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press,
pp. 160-167, 1985.

Gray, A. ``Christoffel Symbols.'' §20.3 in
*Modern Differential Geometry of Curves and Surfaces.* Boca Raton, FL: CRC Press,
pp. 397-400, 1993.

Morse, P. M. and Feshbach, H. *Methods of Theoretical Physics, Part I.*
New York: McGraw-Hill, pp. 47-48, 1953.

© 1996-9

1999-05-26