Frobenius theorem (differential topology)
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.
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 C1 functions, with r < n, and such that the matrix ( f i
k ) has rank r. Consider the following system of partial differential equations for a C2 function u : Rn → R:
One seeks conditions on the existence of a collection of solutions u1, ..., un−r such that the gradients ∇u1, ..., ∇un−r are linearly independent.
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