en>David Eppstein: more start than stub

2013-10-02T06:51:10Z

more start than stub

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	~~Greetings~~. ~~The~~ [~~http~~:/~~/Depts~~.~~Washington~~.~~edu~~/~~handbook~~/~~syndromesBoth~~/~~ch9_ulcers~~.~~html writer~~'s] ~~name std home test~~ is ~~Phebe~~ and ~~she feels comfy when people use~~ [http://~~wmazowiecku~~.pl/~~stay~~-~~yeast~~-~~infection-free-using-these-helpful-suggestions/ over the counter std test]~~ [http://www.~~aseandate~~.~~com~~/~~index~~.~~php?m=member_profile&p=profile&id=13345213 home std test kit~~] ~~complete title~~. ~~Hiring has been my occupation for some time but I~~'~~ve already applied for an additional one~~. ~~North Dakota is her beginning place but she will have to transfer 1 working day or an additional~~. ~~The thing she adores most is body developing and now she is trying to make money with it~~.~~<br><br>Also visit my web site~~: ~~std home test (~~[http://~~facehack~~.ir/~~index.php~~?do=~~/profile~~-~~110/info/ simply click the up coming internet site~~])		In the theory of [[differential form]]s, a '''differential ideal''' ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a [[smooth manifold]], in other words a [[graded algebra\|graded ideal]] in the sense of [[ring theory]], that is further closed under [[exterior differentiation]] ''d''. In other words, for any form α in ''I'', the exterior derivative ''d''α is also in ''I''.

			In the theory of [[differential algebra]], 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''' on a manifold ''M'' is a differential ideal

			:<math> I\subset \Omega^*(M) </math>.

			One can express any [[partial differential equation]] system as an exterior differential system with independence condition. Say that we have ''k''th order partial differential equation systems for maps <math> f: \mathbb{R}^m \rightarrow \mathbb{R}^n</math>, given by

			:<math> F^r(x, u, \frac{\partial^{\|I\|}u }{\partial x^I})=0, \quad 1\le \|I\|\le k </math>.

			The solution of this partial differential equation system is the submanifold <math> \Sigma </math> of the [[jet (mathematics)\|jet space]] consisting of integral manifolds of the pullback of the [[jet bundle\|contact system]] to <math> \Sigma </math>.

			This idea allows one to analyze the properties of partial differential equations with methods of differential geometry. For instance, we can apply [[Cartan's method]] on partial differential equation systems by writing down the exterior differential system associated with it.

			==Perfect differential ideals==
			a differential ideal <math>I \, </math> which has the property that if it contains an element <math> a \in I </math> they contain any element <math> b \in I </math> such that <math> b^n \in I </math> for some <math> n > 0 \, </math> is equal to <math> a \, </math>.

			== References ==
			* Robert Bryant, Phillip Griffiths and Lucas Hsu, [http://www.math.duke.edu/preprints/94-12.dvi ''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

			* Oleg I. Morozov, [http://www.emis.de/journals/SIGMA/2005/Paper006/sigma05-006.pdf ''Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches''](PDF file), SIGMA, Vol. 1 (2005), Paper 006, 14 pages; math-ph/0508016

			*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 URL:[http://links.jstor.org/sici?sici=0002-9947%28193404%2936%3A2%3C361%3AITAADE%3E2.0.CO%3B2-7]

			[[Category:Differential forms]]
			[[Category:Differential algebra]]
			[[Category:Differential systems]]


			{{differential-geometry-stub}}

en>David Eppstein: refimprove

2012-08-24T06:46:54Z

refimprove

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en>David Eppstein: more start than stub

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