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Peterson-Mainardi-Codazzi Equations

Let x:U->R^3 be a regular patch, where U is an open subset of R^2. Then

(partiale)/(partialv)-(partialf)/(partialu)=eGamma_(12)^1+f(Gamma_(12)^2-Gamma_(11)^1)-gGamma_(11)^2
(1)
(partialf)/(partialv)-(partialg)/(partialu)=eGamma_(22)^1+f(Gamma_(22)^2-Gamma_(12)^1)-gGamma_(12)^2,
(2)

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

(partiale)/(partialv)=1/2E_v(e/E+g/G)
(3)
(partialg)/(partialu)=1/2G_u(e/E+g/G)
(4)

(Gray 1997, p. 651), where E, F, and G are coefficients of the first fundamental form.

For an asymptotic patch x:U->R^3,

(partial(lnf))/(partialu)=Gamma_(11)^1-Gamma_(12)^2
(5)
(partial(lnf))/(partialv)=Gamma_(22)^2-Gamma_(12)^1
(6)

and

partial/(partialu)((lnf)/(sqrt(EG-F^2)))=-2Gamma_(12)^2
(7)
partial/(partialv)((lnf)/(sqrt(EG-F^2)))=-2Gamma_(12)^1
(8)

(Gray 1997, p. 660).

See also

Christoffel Symbol of the Second Kind, First Fundamental Form, Second Fundamental Form

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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.

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Peterson-Mainardi-Codazzi Equations

Cite this as:

Weisstein, Eric W. "Peterson-Mainardi-Codazzi Equations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Peterson-Mainardi-CodazziEquations.html

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