Monge equation

In the mathematical theory of partial differential equations, a Monge equation,[1] named after Gaspard Monge, is a first-order partial differential equation for an unknown function u in the independent variables x1,...,xn

F ( u , x 1 , x 2 , … , x n , ∂ u ∂ x 1 , … , ∂ u ∂ x n ) = 0 {\displaystyle 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}

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

∑ i 0 + ⋯ + i n = k P i 0 … i n ( x 0 , x 1 , … , x k ) d x 0 i 0 d x 1 i 1 ⋯ d x n i n = 0 {\displaystyle \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}

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.

References

  1. ^ "Monge method". evlm.stuba.sk. Retrieved 2022-10-16.
  • Rozov, N. Kh. (1990). "Monge equation". In Hazewinkel, Michiel (ed.). Encyclopedia of Mathematics. Vol. 6. Kluwer Academic Publishers. ISBN 1-55608-005-0.


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