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, 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.

Contact geometry studies 1-forms that maximally violates the assumptions of Frobenius' theorem.

One can imagine starting with a cloud of little planes, and quilting them together to form a full surface.

The main danger is that, if we quilt the little planes two at a time, we might go on a cycle and return to where we began, but shifted by a small amount.

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 ik ) has rank r when evaluated at any point of Rn.

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.

The Frobenius theorem asserts that this problem admits a solution locally[3] if, and only if, the operators Lk satisfy a certain integrability condition known as involutivity.

Specifically, they must satisfy relations of the form for 1 ≤ i, j ≤ r, and all C2 functions u, and for some coefficients ckij(x) that are allowed to depend on x.

In other words, the commutators [Li, Lj] must lie in the linear span of the Lk at every point.

In fact, the strategy of proof of the Frobenius theorem is to form linear combinations among the operators Li so that the resulting operators do commute, and then to show that there is a coordinate system yi for which these are precisely the partial derivatives with respect to y1, ..., yr.

Thus, because of the existence of a family of level surfaces, solutions of the original equation are in a one-to-one correspondence with arbitrary functions of one variable.

Frobenius' theorem allows one to establish a similar such correspondence for the more general case of solutions of (1).

Just as in the case of the example, general solutions u of (1) are in a one-to-one correspondence with (continuously differentiable) functions on the family of level sets.

Frobenius' original version of the theorem was stated in terms of Pfaffian systems, which today can be translated into the language of differential forms.

In the vector field formulation, the theorem states that a subbundle of the tangent bundle of a manifold is integrable (or involutive) if and only if it arises from a regular foliation.

, which is a system of first-order ordinary differential equations, whose solvability is guaranteed by the Picard–Lindelöf theorem.

The Frobenius theorem states that F is integrable if and only if for every p in U the stalk Fp is generated by r exact differential forms.

Geometrically, the theorem states that an integrable module of 1-forms of rank r is the same thing as a codimension-r foliation.

The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between differential forms and Lie derivatives.

Frobenius' theorem is one of the basic tools for the study of vector fields and foliations.

Consequently, the Frobenius theorem takes on the equivalent form that I(D) is closed under exterior differentiation if and only if D is integrable.

The conditions of the Frobenius theorem depend on whether the underlying field is R or C. If it is R, then assume F is continuously differentiable.

Let E be a subbundle of the tangent bundle of M. The bundle E is involutive if, for each point p ∈ M and pair of sections X and Y of E defined in a neighborhood of p, the Lie bracket of X and Y evaluated at p, lies in Ep: On the other hand, E is integrable if, for each p ∈ M, there is an immersed submanifold φ : N → M whose image contains p, such that the differential of φ is an isomorphism of TN with φ−1E.

Despite being named for Ferdinand Georg Frobenius, the theorem was first proven by Alfred Clebsch and Feodor Deahna.

Frobenius is responsible for applying the theorem to Pfaffian systems, thus paving the way for its usage in differential topology.

In classical thermodynamics, Frobenius' theorem can be used to construct entropy and temperature in Carathéodory's formalism.

By plugging in the ideal gas laws, and noting that Joule expansion is an (irreversible) adiabatic process, we can fix the sign of