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Frobenius theorem (differential topology) - Wikipedia

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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, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of r vector fields mesh into coordinate grids on r-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds. Contact geometry studies 1-forms that maximally violates the assumptions of Frobenius' theorem. An example is shown on the r

Frobenius theorem (differential topology) - Wikipedia Jump to content From Wikipedia, the free encyclopedia On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs For other theorems named after Frobenius, see Frobenius theorem . The 1-form d z − y d x . on R 3 maximally violates the assumption of Frobenius' theorem. These planes appear to twist along the y -axis. It is not integrable, as can be verified by drawing an infinitesimal square in the x-y plane, and follow the path along the one-forms. The path would not return to the same z-coordinate after one circ

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