Peterson-Mainardi-Codazzi Equations
Let x:U->R^3 be a regular patch, where U is an open subset of R^2. Then
where e, f, and g are coefficients of the second fundamental form and Gamma_(ij)^k are Christoffel symbols of the second kind (Gray 1997, p. 649).
Let x:U->R^3 be a principal patch. Then
(Gray 1997, p. 651), where E, F, and G are coefficients of the first fundamental form.
For an asymptotic patch x:U->R^3,
and
(Gray 1997, p. 660).
Together with the Gauss equation K=(eg-f^2)/(EG-F^2), which expresses the Gaussian curvature K in terms of the coefficients of the first fundamental form and second fundamental form, these equations form the Gauss-Codazzi equations.
See also
Christoffel Symbol of the Second Kind, First Fundamental Form, Gauss-Codazzi Equations, Second Fundamental FormExplore with Wolfram|Alpha
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References
Codazzi, D. "Sulle coordinate curvilinee d'una superficie dello spazio." Ann. math. pura applicata 2, 101-19, 1868-1869.Coolidge, J. L. A History of Geometrical Methods. New York: Dover, 1963.Gray, A. "The Peterson-Mainardi-Codazzi Equations." §28.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 649-652, 1997.Green, A. E. and Zerna, W. Theoretical Elasticity, 2nd ed. New York: Dover, p. 37, 1992.Mainardi, G. "Sulle coordinate curvilinee d'una superfice dello spazio." Giornale del R. Istituto Lombardo 9, 385-398, 1856.Peterson, K. M. "Ueber die Biegung der Flächen." Dorpat. Kandidatenschrift. 1853.Reich, K. "Die Geschichte der Differentialgeometrie von Gauß bis Riemann." Arch. Hist. Exact Sci. 11, 273-382, 1973.Referenced on Wolfram|Alpha
Peterson-Mainardi-Codazzi EquationsCite this as:
Weisstein, Eric W. "Peterson-Mainardi-Codazzi Equations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Peterson-Mainardi-CodazziEquations.html