Monge equation
# 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 $x_1,...,x_n$

    $F\left(u,x_{1},x_{2},\dots ,x_{n},{\frac {\partial u}{\partial x_{1}}},\dots ,{\frac {\partial u}{\partial x_{n}}}\right)=0$

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

Classically, putting u = $x_0,$ a Monge equation of degree k is written in the form

    $\sum _{i_{0}+\cdots +i_{n}=k}P_{i_{0}\dots i_{n}}(x_{0},x_{1},\dots ,x_{k})\,dx_{0}^{i_{0}}\,dx_{1}^{i_{1}}\cdots dx_{n}^{i_{n}}=0$

and expresses a relation between the differentials $dx_k.$ 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.

