Monge cone

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

F(x,y,u,u_x,u_y) = 0\qquad\qquad (1)

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} and 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

z_0 = u(x_0, y_0).\qquad\qquad (2)

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

z = u(x,y)\,

through the point (x0,y0,z0). As the pair (p, q) 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 true cone since a nontrivial and non-coaxial one-parameter family of planes through a fixed point envelopes a cone.)

As the base point (x0,y0,z0) varies, the cone also varies. Thus the Monge cone is a cone field on R3. Finding solutions of (1) can thus be interpreted as finding a surface which is everywhere tangent to the Monge cone at the point. This is the method of characteristics.

The technique generalizes to scalar first-order partial differential equations in n spatial variables; namely,

F\left(x_1,\dots,x_n,u,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right) = 0.

Through each point (x_1^0,\dots,x_n^0, z^0), the Monge cone (or axis in the quasilinear case) is the envelop of solutions of the PDE with u(x_1^0,\dots,x_n^0) = z^0.

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