View source for Differential parameter

''differentiator''

The joint [[Differential invariant|differential invariant]] of one or more functions and the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322701.png" /> of a Riemannian geometry.

The first-order differential parameter (or simply differential parameter) of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322702.png" /> is the square of its gradient:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322703.png" /></td> </tr></table>

The first-order mixed differential parameter of two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322705.png" /> is the scalar product of their gradients

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322706.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322707.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322708.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d0322709.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d03227010.png" /> is the determinant of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d03227011.png" />. In three-dimensional Euclidean space and with respect to a Cartesian rectangular coordinate system, the second differential parameter is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032270/d03227012.png" /></td> </tr></table>

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

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.J. Stoker,   "Differential geometry" , Wiley (Interscience)  (1969)</TD></TR></table>