Monge equation - Alchetron, The Free Social Encyclopedia

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 , … , x n , ∂ u ∂ x 1 , … , ∂ u ∂ 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

∑ i 0 + ⋯ + i n = k P i 0 … i n ( x 0 , x 1 , … , x k ) d x 0 i 0 d x 1 i 1 ⋯ 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

Monge equation Wikipedia

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