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Monge cone
 
 
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In the mathematical
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 theory of partial differential equation
Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
s (PDE), the Monge cone is a geometrical object associated to a first-order equation. It is named for Gaspard Monge
Gaspard Monge

Gaspard Monge, Comte de P?luse , was the inventor of descriptive geometry....
. 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 and 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).






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In the mathematical
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 theory of partial differential equation
Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
s (PDE), the Monge cone is a geometrical object associated to a first-order equation. It is named for Gaspard Monge
Gaspard Monge

Gaspard Monge, Comte de P?luse , was the inventor of descriptive geometry....
. 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 and 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 (p, q) solving (1) varies, the tangent planes envelope
Envelope (mathematics)

In mathematics, an envelope of a index set#Families of manifolds is a manifold that is tangent to each member of the family at some point....
 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
Method of characteristics

In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first order partial differential equation, although more generally the method of characteristics is valid for any hyperbolic partial differential equation....
.

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

Through each point , the Monge cone (or axis in the quasilinear case) is the envelop of solutions of the PDE with .

See also

  • Tangent cone
    Tangent cone

    In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities....