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In the [[mathematics|mathematical]] theory of [[partial differential equations]], a '''Monge equation''', named after [[Gaspard Monge]], is a [[first-order partial differential equation]] for an unknown function ''u'' in the independent variables ''x''<sub>1</sub>,...,''x''<sub>''n''</sub>


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:<math>F\left(u,x_1,x_2,\dots,x_n,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right)=0</math>
 
that is a [[polynomial]] in the partial derivatives of ''u''.   Any Monge equation has a [[Monge cone]].
 
Classically, putting ''u''&nbsp;=&nbsp;''x''<sub>0</sub>, a Monge equation of degree ''k'' is written in the form
 
:<math>\sum_{i_0+\cdots+i_n=k} P_{i_0\dots i_n}(x_0,x_1,\dots,x_k) \, dx_0^{i_0} \, dx_1^{i_1} \cdots dx_n^{i_n}=0</math>
 
and expresses a relation between the [[differential of a function|differentials]] ''dx''<sub>''k''</sub>. The Monge cone at a given point (''x''<sub>0</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>) is the zero locus of the equation in the tangent space at the point.
 
The Monge equation is unrelated to the (second-order) [[Monge–Ampère equation]].
 
[[Category:Partial differential equations]]
 
 
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Revision as of 19:13, 9 January 2014

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In the mathematical theory of partial differential equations, a Monge equation, named after Gaspard Monge, is a first-order partial differential equation for an unknown function u in the independent variables x1,...,xn

that is a polynomial in the partial derivatives of u. Any Monge equation has a Monge cone.

Classically, putting u = x0, a Monge equation of degree k is written in the form

and expresses a relation between the differentials dxk. The Monge cone at a given point (x0, ..., xn) is the zero locus of the equation in the tangent space at the point.

The Monge equation is unrelated to the (second-order) Monge–Ampère equation.


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