|
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 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 :. 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 , given by :. The solution of this partial differential equation system is the submanifold of the jet space consisting of integral manifolds of the pullback of the contact system to . 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「differential ideal」の詳細全文を読む スポンサード リンク
|