In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
theory of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s (PDE), the Monge cone is a geometrical object associated with a first-order equation. It is named for
Gaspard Monge
Gaspard Monge, Comte de Péluse (; 9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. Dur ...
. 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 (''x''
0, ''y''
0, ''z''
0) and consider solution functions ''u'' which have
:
Each solution to (1) satisfying (2) determines the
tangent plane
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the graph
:
through the point
. As the pair (''u''
''x'', ''u''
''y'') solving (1) varies, the tangent planes
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card.
Traditional envelopes are made from sheets of paper cut to one o ...
a cone in R
3 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
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
of ''R''
3, defined at a point (''x'',''y'',''z'') by
:
Vanishing of ''F'' determines a curve in the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
with
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
(''a'':''b'':''c''). The
dual curve is a curve in the projective
tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
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 R
3. 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 (, methodos, from μετά/meta "in pursuit or quest of" + ὁδός/hodos "a method, system; a way or manner" of doing, saying, etc.), literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In re ...
.
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
.
Examples
;Eikonal equation
The simplest fully nonlinear equation is the
eikonal equation. This has the form
:
so that the function ''F'' is given by
:
The dual cone consists of 1-forms ''a dx'' + ''b dy'' + ''c dz'' satisfying
:
Taken projectively, this defines a circle. The dual curve is also a circle, and so the Monge cone at each point is a proper cone.
See also
*
Tangent cone
References
*
*
* {{cite book, author=Monge, G., title=Application de l'analyse à la géométrie, url=https://archive.org/details/bub_gb_iCEOAAAAQAAJ, publisher=Bachelier, year=1850, language=fr
Partial differential equations