Frobenius Theorem (differential Topology)

Frobenius Theorem (differential Topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined 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.