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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 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

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rdfs:label	Monge cone (en)
rdfs:comment	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 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 (en)
dcterms:subject	Partial differential equations
Wikipage page ID	15877576 (xsd:integer)
Wikipage revision ID	1018562561 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective plane David Hilbert Richard Courant Mathematics Eikonal equation Envelope (mathematics) Gaspard Monge Tangent space Dual curve Partial differential equation Cotangent bundle Partial differential equations Homogeneous coordinates Differential equations Method of characteristics Tangent cone Tangent plane
Link from a Wikipage to an external page	https://archive.org/details/bub_gb_iCEOAAAAQAAJ%7Cpublisher=Bachelier%7Cyear=1850%7Clanguage=fr
sameAs	Monge cone Monge cone Monge cone Monge cone
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Cite_book
author	Ivanov, A.B. (en)
id	m/m064630 (en)
title	Monge cone (en)
has abstract	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 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 . As the pair (ux, uy) solving (1) varies, the tangent planes envelope a cone in R3 with vertex at , 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 Vanishing of F determines a curve in the projective plane with homogeneous coordinates (a:b:c). The dual curve is a curve in the projective tangent space at the point, and the affine cone over this curve is the Monge cone. The cone may have multiple branches, each one an affine cone over a simple closed curve in the projective tangent space. As the base point 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, Through each point , the Monge cone (or axis in the quasilinear case) is the envelope of solutions of the PDE with . (en)
prov:wasDerivedFrom	wikipedia-en:Monge_cone?oldid=1018562561&ns=0
page length (characters) of wiki page	3753 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Monge_cone
is Link from a Wikipage to another Wikipage of	Light cone Envelope (mathematics) Gaspard Monge Tube domain Method of characteristics Monge equation Tangent cone Characteristic cone Monge axis
is Wikipage redirect of	Characteristic cone Monge axis
is foaf:primaryTopic of	wikipedia-en:Monge_cone

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