In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Cartan–Kähler theorem is a major result on the
integrability conditions for differential systems In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
, in the case of
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s, for
differential ideal In the theory of differential forms, a differential ideal ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exter ...
s
. It is named for
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
and
Erich Kähler
Erich Kähler (; 16 January 1906 – 31 May 2000) was a German mathematician with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory.
Education an ...
.
Meaning
It is not true that merely having
contained in
is sufficient for integrability. There is a problem caused by
singular solution A singular solution ''ys''(''x'') of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the so ...
s. The theorem computes certain constants that must satisfy an inequality in order that there be a solution.
Statement
Let
be a real analytic
EDS. Assume that
is a connected, ''
''-dimensional, real analytic, regular
integral manifold In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of ...
of ''
'' with
(i.e., the tangent spaces
are "extendable" to higher dimensional integral elements).
Moreover, assume there is a real analytic submanifold
of codimension
containing
and such that
has dimension
for all
.
Then there exists a (locally) unique connected,
-dimensional, real analytic integral manifold
of
that satisfies
.
Proof and assumptions
The
Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.
References
*
Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymo ...
, ''Eléments d'analyse'', vol. 4, (1977) Chapt. XVIII.13
*R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, ''Exterior Differential Systems'', Springer Verlag, New York, 1991.
External links
*
*R. Bryant
"Nine Lectures on Exterior Differential Systems" 1999
E. Cartan, "On the integration of systems of total differential equations," transl. by D. H. DelphenichE. Kähler, "Introduction to the theory of systems of differential equations," transl. by D. H. Delphenich
{{DEFAULTSORT:Cartan-Kahler theorem
Partial differential equations
Theorems in analysis