Monge Equation

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 ''x''₁,...,''x''_''n''

:$F\left(u,x_1,x_2,\dots,x_n,\frac,\dots,\frac\right)=0$

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

Classically, putting ''u'' = ''x''₀, a Monge equation of degree ''k'' is written in the form

:$\sum_ P_(x_0,x_1,\dots,x_k) \, dx_0^ \, dx_1^ \cdots dx_n^=0$

and expresses a relation between the differentials ''dx''_''k''. The Monge cone at a given point (''x''₀, ..., ''x''_''n'') 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.

References

*

Partial differential equations

{{mathanalysis-stub