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 x₁,...,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 = x₀, 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 (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.