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Gauss Equations


If x is a regular patch on a regular surface in R^3 with normal N^^, then

x_(uu)=Gamma_(11)^1x_u+Gamma_(11)^2x_v+eN^^
(1)
x_(uv)=Gamma_(12)^1x_u+Gamma_(12)^2x_v+fN^^
(2)
x_(vv)=Gamma_(22)^1x_u+Gamma_(22)^2x_v+gN^^,
(3)

where e, f, and g are coefficients of the second fundamental form and Gamma_(ij)^k are Christoffel symbols of the second kind.


See also

Christoffel Symbol of the Second Kind, Fundamental Forms, Peterson-Mainardi-Codazzi Equations

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 511-512, 1997.

Referenced on Wolfram|Alpha

Gauss Equations

Cite this as:

Weisstein, Eric W. "Gauss Equations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussEquations.html

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