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 $u(x_1^0,\dots,x_n^0) = z^0$ .

[edit] See also

Tangent cone

[edit] References

David Hilbert and Richard Courant (1989). Methods of mathematical physics, Volume 2. Wiley Interscience.
Ivanov, A.B. (2001), “Monge cone”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Monge, G. (1850). Application de l'analyse à la géométrie. Bachelier. (French)