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In the mathematical theory of partial differential equations , 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 F u x {\displaystyle F_{u_{x}}} and F u y {\displaystyle F_{u_{y}}} are not both zero in the domain of definition. Fix a point and consider solution functions u which have
Each solution to satisfying determines the tangent plane to the graph
through the point x 0 , y 0 , z 0 {\displaystyle x_{0},y_{0},z_{0}}. As the pair solving varies, the tangent planes envelope a cone in R with vertex at x 0 , y 0 , z 0 {\displaystyle x_{0},y_{0},z_{0}} , 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 R, defined at a point by