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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 mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, in other words a graded ideal in the sense of
ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, that is further closed under exterior differentiation ''d'', meaning that for any form α in ''I'', the exterior derivative ''d''α is also in ''I''. In the theory of
differential algebra In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A n ...
, a differential ideal ''I'' in a differential ring ''R'' is an ideal which is mapped to itself by each differential operator.


Exterior differential systems and partial differential equations

An exterior differential system consists of a smooth manifold M and a differential ideal : I\subset \Omega^*(M) . An integral manifold of an exterior differential system (M,I) consists of a
submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
N\subset M having the property that the pullback to N of all differential forms contained in I vanishes identically. One can express any partial differential equation system as an exterior differential system with independence condition. Suppose that we have a ''k''th order partial differential equation system for maps u: \mathbb^m \rightarrow \mathbb^n, given by : F^r(x, u, \frac)=0, \quad 1\le , I, \le k . The graph of the k-jet (u^a,p^a_i,\dots,p^a_I)=(u^a(x),\frac,\dots,\frac)_ of any solution of this partial differential equation system is a submanifold N of the
jet space In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...
, and is an integral manifold of the
contact system In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ...
du^a-p^a_i dx^i,\dots,dp^a_I-p^p_ dx^j_on the k-jet bundle. This idea allows one to analyze the properties of partial differential equations with methods of differential geometry. For instance, we can apply the
Cartan–Kähler_theorem In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I. It is named for Élie Cartan and Erich Kähler. Meaning It is no ...
to a system of partial differential equations by writing down the associated exterior differential system. We can frequently apply Cartan's equivalence method to exterior differential systems to study their symmetries and their diffeomorphism invariants.


Perfect differential ideals

A differential ideal I \, is perfect if it has the property that if it contains an element a \in I then it contains any element b \in I such that b^n = a for some n > 0 \, .


References

* Robert Bryant,
Phillip Griffiths Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...
and Lucas Hsu
''Toward a geometry of differential equations''
DVI file), in Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topology, edited by S.-T. Yau, vol. IV (1995), pp. 1–76, Internat. Press, Cambridge, MA * Robert Bryant,
Shiing-Shen Chern Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...
, Robert Gardner,
Phillip Griffiths Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...
, Hubert Goldschmidt, Exterior Differential Systems, Springer--Verlag, Heidelberg, 1991. *Thomas A. Ivey, J. M. Landsberg, Cartan for beginners. Differential geometry via moving frames and exterior differential systems. Second edition. Graduate Studies in Mathematics, 175. American Mathematical Society, Providence, RI, 2016. *H. W. Raudenbush, Jr. "Ideal Theory and Algebraic Differential Equations", ''Transactions of the American Mathematical Society'', Vol. 36, No. 2. (Apr., 1934), pp. 361–368. Stable UR

*J. F. Ritt, Differential Algebra, Dover, New York, 1950. Differential forms Differential algebra Differential systems {{differential-geometry-stub