Cayley-Darboux equation

A third-order partial differential equation that must be necessarily satisfied by a function $ u ( x _ {1} , x _ {2} , x _ {3} ) $ in order for the family of surfaces $ u ( x _ {1} , x _ {2} , x _ {3} ) = \textrm{ const } $ to be complementable to a triply orthogonal system of surfaces. The Cayley–Darboux equation may be written as

$$ \left | \begin{array}{cccccc} c _ {11} &c _ {22} &c _ {33} &2c _ {12} &2c _ {23} &2c _ {31} \\ u _ {11} &u _ {22} &u _ {33} &2u _ {12} &2u _ {23} &2u _ {31} \\ 1 & 1 & 1 & 0 & 0 & 0 \\ u _ {1} & 0 & 0 &u _ {2} & 0 &u _ {3} \\ 0 &u _ {2} & 0 &u _ {1} &u _ {3} & 0 \\ 0 & 0 &u _ {3} & 0 &u _ {2} &u _ {1} \\ \end{array} \ \right | = 0, $$

where

$$ c _ {\alpha \beta } = \ \sum _ {k = 1 } ^ { 3 } ( u _ {k} u _ {\alpha \beta k } - 2u _ {\alpha k } u _ {\beta k } ), $$

and

$$ u _ {k} = \ u _ {x _ {k} } \dots u _ {\alpha \beta \gamma } = \ u _ {x _ \alpha x _ \beta x _ \gamma } . $$

The equation was first obtained in explicit form by A. Cayley [1]. The above form of the equation is due to G. Darboux [2].

References

[1]	A. Cayley, "Sur la condition pour qu'une famille de surfaces données puisse faire partie d'un système orthogonal" C.R. Acad. Sci. Paris , 75 (1872) pp. 324–330; 381–385 (Also: Collected mathematical papers, Vol. 8 (1891), pp. 269–291)
[2]	G. Darboux, "Leçons sur les systèmes orthogonaux et les coordonnées curvilignes" , Paris (1898)
[3]	V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1948) (In Russian)