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Difference between revisions of "Cayley-Darboux equation"

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A third-order partial differential equation that must be necessarily satisfied by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021030/c0210301.png" /> in order for the family of surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021030/c0210302.png" /> to be complementable to a triply orthogonal system of surfaces. The Cayley–Darboux equation may be written as
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<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/c/c021/c021030/c0210303.png" /></td> </tr></table>
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A third-order partial differential equation that must be necessarily satisfied by a function  $  u ( x _ {1} , x _ {2} , x _ {3} ) $
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in order for the family of surfaces  $  u ( x _ {1} , x _ {2} , x _ {3} ) = \textrm{ const } $
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to be complementable to a triply orthogonal system of surfaces. The Cayley–Darboux equation may be written as
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$$
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\left |
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\begin{array}{cccccc}
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c _ {11}  &c _ {22}  &c _ {33}  &2c _ {12}  &2c _ {23}  &2c _ {31}  \\
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u _ {11}  &u _ {22}  &u _ {33}  &2u _ {12}  &2u _ {23}  &2u _ {31}  \\
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1  & 1  & 1  & 0  & 0  & 0  \\
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u _ {1}  & 0  & 0  &u _ {2}  & 0  &u _ {3}  \\
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0  &u _ {2}  & 0  &u _ {1}  &u _ {3}  & 0  \\
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0  & 0  &u _ {3}  & 0  &u _ {2}  &u _ {1}  \\
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\end{array}
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\
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\right |  = 0,
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$$
  
 
where
 
where
  
<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/c/c021/c021030/c0210304.png" /></td> </tr></table>
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$$
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c _ {\alpha \beta }  = \
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\sum _ {k = 1 } ^ { 3 }
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( u _ {k} u _ {\alpha \beta k }  -
 +
2u _ {\alpha k }  u _ {\beta k }  ),
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$$
  
 
and
 
and
  
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$$
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u _ {k}  = \
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u _ {x _ {k}  } \dots
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u _ {\alpha \beta \gamma }  = \
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u _ {x _  \alpha  x _  \beta  x _  \gamma  } .
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$$
  
 
The equation was first obtained in explicit form by A. Cayley [[#References|[1]]]. The above form of the equation is due to G. Darboux [[#References|[2]]].
 
The equation was first obtained in explicit form by A. Cayley [[#References|[1]]]. The above form of the equation is due to G. Darboux [[#References|[2]]].

Latest revision as of 16:43, 4 June 2020


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)
How to Cite This Entry:
Cayley-Darboux equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley-Darboux_equation&oldid=46285
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article