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 EquationsExplore with Wolfram|Alpha
WolframAlpha
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 EquationsCite this as:
Weisstein, Eric W. "Gauss Equations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GaussEquations.html