Differential parameter

differentiator

The joint differential invariant of one or more functions and the metric tensor $g_{ij}$ of a Riemannian geometry.

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

$$\Delta_1V=g^{ij}V_iV_j.$$

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

$$\Delta_1(V,W)=g^{ij}V_iW_j.$$

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

$$\Delta_1(V)=\left(\frac{\partial V}{\partial x}\right)^2+\left(\frac{\partial V}{\partial y}\right)^2+\left(\frac{\partial V}{\partial z}\right)^2,$$

$$\Delta_1(V,W)=\frac{\partial V}{\partial x}\frac{\partial W}{\partial x}+\frac{\partial V}{\partial y}\frac{\partial W}{\partial y}+\frac{\partial V}{\partial z}\frac{\partial W}{\partial z}.$$

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

$$\Delta_2(V)=g^{ij}\nabla_iV_j=\frac{1}{\sqrt g}\frac{\partial}{\partial x^i}(\sqrt{gg}^{ij}V_j),$$

where $g$ is the determinant of the matrix $\|g_{ij}\|$. In three-dimensional Euclidean space and with respect to a Cartesian rectangular coordinate system, the second differential parameter is given by the formula

$$\Delta_2(V)=\frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}+\frac{\partial^2V}{\partial z^2}.$$

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)