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

$F(u,x_1,x_2,\dots ,x_n,\frac{\partial u}{\partial x_1},\dots ,\frac{\partial u}{\partial x_n})=0$

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

$\sum _{i_0+\cdots +i_n=k}{P}_{i_0\dots i_n}(x_0,x_1,\dots ,x_k)d{x}_0^{i_0}d{x}_1^{i_1}\cdots d{x}_n^{i_n}=0$

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.

References

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