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Difference between revisions of "Monge equation"

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A differential equation of the form
 
A differential equation of the form
  
<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/m/m064/m064640/m0646401.png" /></td> </tr></table>
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$$F(x,y,z,dx,dy,dz)=0.$$
  
 
G. Monge (see [[#References|[1]]]) studied these equations in connection with the construction of a geometric theory of first-order partial differential equations. A particular case of a Monge equation is a [[Pfaffian equation|Pfaffian equation]].
 
G. Monge (see [[#References|[1]]]) studied these equations in connection with the construction of a geometric theory of first-order partial differential equations. A particular case of a Monge equation is a [[Pfaffian equation|Pfaffian equation]].
  
For example, consider a first-order partial differential equation for an unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064640/m0646402.png" /> of two independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064640/m0646403.png" />:
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For example, consider a first-order partial differential equation for an unknown function $z$ of two independent variables $x,y$:
  
<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/m/m064/m064640/m0646404.png" /></td> </tr></table>
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$$\Phi\left(x,y,z,\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right)=0,$$
  
 
then the directions of the generators of the [[Monge cone|Monge cone]] (characteristic directions) at each point have to satisfy the Monge equation, which can be written in the form
 
then the directions of the generators of the [[Monge cone|Monge cone]] (characteristic directions) at each point have to satisfy the Monge equation, which can be written in the form
  
<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/m/m064/m064640/m0646405.png" /></td> </tr></table>
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$$M\left(x,y,z,\frac{dy}{dz},\frac{dz}{dx}\right)=0.$$
  
 
This is an ordinary differential equation in two unknown functions, that is, it is the simplest case of an [[Underdetermined system|underdetermined system]]. Often an arbitrary underdetermined system of ordinary differential equations, in which the number of equations is less than the number of unknown functions, is called a Monge equation.
 
This is an ordinary differential equation in two unknown functions, that is, it is the simplest case of an [[Underdetermined system|underdetermined system]]. Often an arbitrary underdetermined system of ordinary differential equations, in which the number of equations is less than the number of unknown functions, is called a Monge equation.

Latest revision as of 09:01, 1 August 2014

A differential equation of the form

$$F(x,y,z,dx,dy,dz)=0.$$

G. Monge (see [1]) studied these equations in connection with the construction of a geometric theory of first-order partial differential equations. A particular case of a Monge equation is a Pfaffian equation.

For example, consider a first-order partial differential equation for an unknown function $z$ of two independent variables $x,y$:

$$\Phi\left(x,y,z,\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right)=0,$$

then the directions of the generators of the Monge cone (characteristic directions) at each point have to satisfy the Monge equation, which can be written in the form

$$M\left(x,y,z,\frac{dy}{dz},\frac{dz}{dx}\right)=0.$$

This is an ordinary differential equation in two unknown functions, that is, it is the simplest case of an underdetermined system. Often an arbitrary underdetermined system of ordinary differential equations, in which the number of equations is less than the number of unknown functions, is called a Monge equation.

References

[1] G. Monge, "Application de l'analyse à la géométrie" , Bachelier (1850)
[2] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[3] P.K. Rashevskii, "Geometric theory of partial differential equations" , Moscow-Leningrad (1947) (In Russian)
How to Cite This Entry:
Monge equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monge_equation&oldid=32639
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article