Gauss Equations

If x is a regular patch on a Regular Surface in $\Bbb{R}^3$ with normal $\hat{\bf N}$, then

$${\bf x}_{uu} = \Gamma_{11}^1 {\bf x}_u+\Gamma_{11}^2 {\bf x}_v+e\hat{\bf N}$$ (1)

$${\bf x}_{uv} = \Gamma_{12}^1 {\bf x}_u+\Gamma_{12}^2 {\bf x}_v+f\hat{\bf N}$$ (2)

$${\bf x}_{vv} = \Gamma_{22}^1 {\bf x}_u+\Gamma_{22}^2 {\bf x}_v+g\hat{\bf N},$$ (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, Mainardi-Codazzi Equations

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 398-400, 1993.