Mainardi-Codazzi Equations

$${\partial e\over\partial v}-{\partial f\over\partial u} = e\Gamma_{12}^1+f(\Gamma_{12}^2-\Gamma_{11}^1)-g\Gamma_{11}^2$$ (1)

$${\partial f\over\partial v}-{\partial g\over\partial u} = e\Gamma_{22}^1+f(\Gamma_{22}^2-\Gamma_{12}^1)-g\Gamma_{12}^2,$$ (2)

where $e$, $f$, and $g$ are coefficients of the second Fundamental Form and $\Gamma_{ij}^k$ are Christoffel Symbols of the Second Kind. Therefore,

$${\partial e\over\partial v} = {\textstyle{1\over 2}}E_v\left({{e\over E}+{g\over G}}\right)$$ (3)

$${\partial g\over\partial u} = {\textstyle{1\over 2}}G_u\left({{e\over E}+{g\over G}}\right)$$ (4)

$${\partial(\ln f)\over\partial u} = \Gamma_{11}^1-\Gamma_{12}^2$$ (5)

$${\partial(\ln f)\over\partial v} = \Gamma_{22}^2-\Gamma_{12}^1$$ (6)

$${\partial\over\partial u}\left({\ln f\over\sqrt{EG-F^2}}\right) = -2\Gamma_{12}^2$$ (7)

$${\partial\over\partial v}\left({\ln f\over\sqrt{EG-F^2}}\right) = -2\Gamma_{12}^1,$$ (8)

where $E$, $F$, and $G$ are coefficients of the first Fundamental Form.

References

Gray, A. ``The Mainardi-Codazzi Equations.'' §20.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 401-402, 1993.