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.