Monge equation explained

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,...,x_n

F\left(u,x _{1,x} _{2,...,x} , ... , _{n, (\partial u)/(\partial x _{1})} (\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

\sum _{i _{0+ … +i} _{n=k}} P _{i _{0...} i _{n}} (x _{0,x} _{1,...,x} _{k)} dx ^{i _{0}} _{0} dx ^{i _{1}} _{1} … dx ^{i _{n}} _{n} =0

and expresses a relation between the differentials dx_k. The Monge cone at a given point (x₀, ..., 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

• Book: Rozov, N. Kh.. Encyclopedia of Mathematics. Kluwer Academic Publishers. 1990. 1-55608-005-0. Hazewinkel. Michiel. 6. Monge equation.

Notes and References

1. Web site: Monge method . 2022-10-16 . evlm.stuba.sk.