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,...,xn
\( {\displaystyle F\left(u,x_{1},x_{2},\dots ,x_{n},{\frac {\partial u}{\partial x_{1}}},\dots ,{\frac {\partial u}{\partial x_{n}}}\right)=0} \)
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
\({\displaystyle \sum _{i_{0}+\cdots +i_{n}=k}P_{i_{0}\dots i_{n}}(x_{0},x_{1},\dots ,x_{k})\,dx_{0}^{i_{0}}\,dx_{1}^{i_{1}}\cdots dx_{n}^{i_{n}}=0} \)
and expresses a relation between the differentials dxk. The Monge cone at a given point (x0, ..., xn) 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.
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