Differential parameter

differentiator

The joint differential invariant of one or more functions and the metric tensor of a Riemannian geometry.

The first-order differential parameter (or simply differential parameter) of a function is the square of its gradient:

The first-order mixed differential parameter of two functions and is the scalar product of their gradients

In three-dimensional Euclidean space and with respect to a Cartesian rectangular coordinate system these differential parameters are given by the formulas

The second-order differential parameter (or second differential parameter) of a function is the divergence of its gradient:

where is the determinant of the matrix . In three-dimensional Euclidean space and with respect to a Cartesian rectangular coordinate system, the second differential parameter is given by the formula

Differential parameters were first introduced by G. Lamé [1] in Euclidean geometry. E. Beltrami [2] must be credited with generalizing this concept. The differential parameters are therefore sometimes called Lamé or Beltrami differential parameters.

References

[1]	G. Lamé, "Leçons sur les coordonnées curvilignes et leurs diverses applications" , Paris (1958)
[2]	E. Betrami, "Ricerche di analisi applicate alla geometria" G. Mat. Battaglini , 2–3 (1864–1865)
[3]	V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 1–2 , Moscow-Leningrad (1947–1948) (In Russian)
[4]	V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)

References

[a1]	J.J. Stoker, "Differential geometry" , Wiley (Interscience) (1969)