In classical differential geometry of surfaces, the Codazzi-Mainardi equations are expressed via the second fundamental form
Consider a parametric surface in Euclidean space,
where the three component functions depend smoothly on ordered pairs in some open domain in the -plane. Assume that this surface is regular, meaning that the vectors and are linearly independent. Complete this to a basis , by selecting a unit vector normal to the surface. The unit vector is nothing but
It is possible to express the second partial derivatives of using the Christoffel symbols and the second fundamental form.
Clairaut’s theorem states that partial derivatives commute
If we differentiate with respect to and with respect to , we get
Now substitute the above expressions for the second derivatives and equate the coefficients of
Rearranging this equation gives the first Codazzi equation. The second equation may be derived similarly.