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In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields (satisfying an integrability condition) in much the same way as an integral curve may be assigned to a single vector field. The theorem is foundational in differential topology and calculus on manifolds. ==Introduction== In its most elementary form, the theorem addresses the problem of finding a maximal set of independent solutions of a regular system of first-order linear homogeneous partial differential equations. Let : be a collection of functions, with , and such that the matrix has rank ''r''. Consider the following system of partial differential equations for a function : : One seeks conditions on the existence of a collection of solutions such that the gradients are linearly independent. The Frobenius theorem asserts that this problem admits a solution locally〔Here ''locally'' means inside small enough open subsets of . Henceforth, when we speak of a solution, we mean a local solution.〕 if, and only if, the operators satisfy a certain integrability condition known as ''involutivity''. Specifically, they must satisfy relations of the form : for , and all functions ''u'', and for some coefficients ''c''''k''''ij''(''x'') that are allowed to depend on ''x''. In other words, the commutators must lie in the linear span of the at every point. The involutivity condition is a generalization of the commutativity of partial derivatives. In fact, the strategy of proof of the Frobenius theorem is to form linear combinations among the operators so that the resulting operators do commute, and then to show that there is a coordinate system for which these are precisely the partial derivatives with respect to . 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「Frobenius theorem (differential topology)」の詳細全文を読む スポンサード リンク
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