## Christoffel Symbol of the Second Kind

Variously denoted or .

 (1)

where is a Connection Coefficient and is a Christoffel Symbol of the First Kind.
 (2)

The Christoffel symbols are given in terms of the first Fundamental Form , , and by
 (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)

See also Christoffel Symbol of the First Kind, Connection Coefficient, Gauss Equations

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.