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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 way a differential form ''restricts'' to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. Given a collection of differential 1-forms α''i'', ''i''=1,2, ..., ''k'' on an ''n''-dimensional manifold ''M'', an integral manifold is a submanifold whose tangent space at every point ''p'' ∈ ''M'' is annihilated by each α''i''. A maximal integral manifold is a submanifold : such that the kernel of the restriction map on forms : is spanned by the α''i'' at every point ''p'' of ''N''. If in addition the α''i'' are linearly independent, then ''N'' is (''n'' − ''k'')-dimensional. Note that ''i'': ''N'' ⊂ ''M'' need not be an embedded submanifold. A Pfaffian system is said to be completely integrable if ''N'' admits a foliation by maximal integral manifolds. (Note that the foliation need not be regular; i.e. the leaves of the foliation might not be embedded submanifolds.) An integrability condition is a condition on the α''i'' to guarantee that there will be integral submanifolds of sufficiently high dimension. ==Necessary and sufficient conditions== The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal algebraically generated by the collection of α''i'' inside the ring Ω(''M'') is differentially closed, in other words : then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.) 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「Integrability conditions for differential systems」の詳細全文を読む スポンサード リンク
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