Mainardi-Codazzi equations

The Mainardi-Codazzi equations , named after the Italian mathematicians Gaspare Mainardi and Delfino Codazzi , are formulas of classical differential geometry that refer to surfaces in three-dimensional space ( ). They describe a relationship between the coefficient , , the second fundamental form , the partial derivatives according to the description of the surface parameters used and as well as the Christoffel symbols . These equations are also necessary integrability conditions for the Gauß-Weingarten equations . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle L}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle \Gamma _{jk}^{i}}$

${\displaystyle L_{v}-M_{u}=L\Gamma _{12}^{1}+M(\Gamma _{12}^{2}-\Gamma _{11}^{1})-N\Gamma _{11}^{2}}$

${\displaystyle M_{v}-N_{u}=L\Gamma _{22}^{1}+M(\Gamma _{22}^{2}-\Gamma _{12}^{1})-N\Gamma _{12}^{2}}$

Web links

• Eric W. Weisstein : Peterson-Mainardi-Codazzi Equations . In: MathWorld (English).