Monge cone

In the mathematical theory of partial differential equations (PDE), the Monge cone is a geometrical object associated with a first-order equation. It is named for Gaspard Monge. In two dimensions, let

be a PDE for an unknown real-valued function u in two variables x and y. Assume that this PDE is non-degenerate in the sense that ${\displaystyle F_{u_{x}}}$ and ${\displaystyle F_{u_{y}}}$ are not both zero in the domain of definition. Fix a point (x₀, y₀, z₀) and consider solution functions u which have

Each solution to (1) satisfying (2) determines the tangent plane to the graph

through the point (x₀,y₀,z₀). As the pair (u_x, u_y) solving (1) varies, the tangent planes envelope a cone in R³ with vertex at (x₀,y₀,z₀), called the Monge cone. When F is quasilinear, the Monge cone degenerates to a single line called the Monge axis. Otherwise, the Monge cone is a proper cone since a nontrivial and non-coaxial one-parameter family of planes through a fixed point envelopes a cone. Explicitly, the original partial differential equation gives rise to a scalar-valued function on the cotangent bundle of R³, defined at a point (x,y,z) by

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