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 a modern geometric point of view, the theorem assigns an integral manifold to a 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 because of its connection with the theory of foliations. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Michael Spivaks Calculus on Manifolds is a text treating analysis in several variables in Euclidean spaces and on differentiable manifolds. ... In mathematics, a foliation is a geometric device used to study 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. Suppose that f_kⁱ(x) are a collection of real-valued C¹ functions on Rⁿ, for i=1,2,...,n, and k=1,2,...,r, where r<n, such that the matrix (f_kⁱ) has rank r. Consider the following system of partial differential equations for a real-valued C² function u on Rⁿ: In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...

(1)

One seeks conditions on the existence of a collection of solutions u₁, ..., u_n-r such that the gradients

are linearly independent. In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

The Frobenius theorem asserts that this problem admits a solution locally^[1] if, and only if, the operators L_k satisfy a certain integrability condition known as involutivity. Specifically, they must satisfy relations of the form In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ...

for i, j = 1,2,...,r, and all C² functions u, and for some coefficients c^k_ij(x) that are allowed to depend on x.

In other words, the commutators [L_i,L_j] must lie in the linear span of the L_k 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 L_i so that the resulting operators do commute, and then to show that there is a coordinate system y_i for which these are precisely the partial derivatives with respect to y₁, ..., y_r. In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

From analysis to geometry

Solutions to undetermined systems of equations are seldom unique. For example, the system

clearly lacks a unique solution. Nevertheless, the solutions still have enough structure that they may be completely described. The first observation is that, even if f₁ and f₂ are two different solutions, the level surfaces of f₁ and f₂ must overlap. In fact, the level surfaces for this system are all planes in R³ of the form x - y + z = C, for C a constant. The second observation is that, once the level surfaces are known, all solutions can then be given in terms of an arbitrary function. Since the value of a solution f on a level surface is constant by definition, define a function C(t) by: In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ...

f(x,y,z) = C(t) whenever x - y + z = t.

Conversely, if a function C(t) is given, then each function f given by this expression is a solution of the original equation. 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). Suppose that u₁,...,u_n-r are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the level sets^[2] of (u₁,...,u_n-r) regarded as an R^n-r-valued function. If v₁,...,v_n-r is any other such collection of solutions, one can show (using some linear algebra and the mean value theorem) that this has the same family of level sets as the u's, but with a possibly different choice of constants for each set. Thus, even though the independent solutions of (1) are not unique, the equation (1) nonetheless determines a unique family of level sets. 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.^[3] In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ...

The level sets corresponding to the maximal independent solution sets of (1) are called the integral manifolds because functions on the collection of all integral manifolds correspond in some sense to "constants" of integration. Once one of these "constants" of integration is known, then the corresponding solution is also known.

Frobenius' theorem in modern language

The Frobenius theorem can be restated more economically in modern language. Frobenius' original version of the theorem was stated in terms of Pfaffian systems, which today can be translated into the language of differential forms. An alternative formulation, which is somewhat more intuitive, uses vector fields. In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

Formulation using vector fields

In the vector field formulation, the theorem states that a subbundle of the tangent bundle of a manifold is integrable if and only if it arises from a regular foliation. In this context, the Frobenius theorem relates integrability to foliation; to state the theorem, both concepts must be clearly defined. In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Integrability is a mathematical concept used in different areas. ... This article or section does not cite any references or sources. ...

One begins by noting that an arbitrary smooth vector field X on a manifold M can be integrated to define a one-parameter family of curves. The integrability follows because the equation defining the curve is a first-order ordinary differential equation, and thus its integrability is guaranteed by the Picard-Lindelöf theorem. Indeed, vector fields are often defined to be the derivatives of a collection of smooth curves. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, the Picard-Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem has exactly one solution if f is Lipschitz continuous in , continuous in as long as stays bounded. ...

This idea of integrability can be extended to collections of vector fields as well. One says that a subbundle of the tangent bundle TM is integrable, if, for any two vector fields X and Y taking values in E, then the Lie bracket [X,Y] takes values in E as well. This notion of integrability need only be defined locally; that is, the existence of the vector fields X and Y and their integrability need only be defined on subsets of M. In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

A subbundle may also be defined to arise from a foliation of a manifold. Let be a submanifold that is a leaf of a foliation. Consider the tangent bundle TN. If TN is exactly E restricted to N, then one says that E arises from a regular foliation of M. Again, this definition is purely local: the foliation is defined only on charts. In mathematics, a foliation is a geometric device used to study manifolds. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

Given the above definitions, Frobenius' theorem states that a subbundle E is integrable if and only if it arises from a regular foliation of M.

Differential forms formulation

Let U be an open set in a manifold M, Ω¹(U) be the space of smooth, differentiable 1-forms on U, and F be a submodule of Ω¹(U) of rank r, the rank being constant in value over U. The Frobenius theorem states that F is integrable if and only if for every the stalk F_p is generated by r exact differential forms. (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In abstract algebra, a module is a generalization of a vector space. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations d&#945; = 0 for a given form &#945; to be a closed form, and &#945; = d&#946; for an exact form, with &#945; given...

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. In mathematics, a foliation is a geometric device used to study manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

Generalizations

• The statement remains true for holomorphic 1-forms on complex manifolds -- manifolds over C with holomorphic transition functions.
• The statement does not generalize to higher degree forms, although there are a number of partial results such as Darboux's theorem and the Cartan-Kähler theorem.

In mathematics, a complex form is a differential form on a complex manifold. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, a transition function has several different meanings: In topology, a transition function is a homeomorphism from one coordinate chart to another. ... Darbouxs theorem is a theorem in symplectic topology which states that every symplectic manifold (of fixed dimension) is locally symplectomorphic. ... In mathematics, the Cartan-Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I. It is named for Élie Cartan and Erich Kähler It is not true that merely having dI contained in I is...

History

Despite being named for Ferdinand Georg Frobenius, the theorem was first proven by Alfred Clebsch and F. Deahna. Deahna was the first to establish the sufficient conditions for the theorem, and Clebsch developed the necessary conditions. Frobenius is responsible for applying the theorem to Pfaffian systems, thus paving the way for its usage in differential topology. Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ... Alfred Clebsch (1832-1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. ... Heinrich Wilhelm Feodor Deahna (1815-1844) was a German mathematician, best known for providing proof of what is now known as Frobenius theorem (differential topology) in 1840. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ...

Notes

1. ^ Here locally means inside small enough open subsets of Rⁿ. Henceforth, when we speak of a solution, we mean a local solution.
2. ^ A level set is a subset of Rⁿ corresponding to the locus of:

   (u₁,...,u_n-r) = (c₁,...,c_n-r), for some constants c_i.

3. ^ The notion of a continuously differentiable function on a family of level sets can be made rigorous by means of the implicit function theorem.

In multivariable calculus, a branch of mathematics, the implicit function theorem is a tool which allows relations to be converted to functions. ...

See also

• Integrability conditions for differential systems

In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ...

References

• H. B. Lawson, The Qualitative Theory of Foliations, (1977) American Mathematical Society CBMS Series volume 27, AMS, Providence RI.
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See theorem 2.2.26.
• Clebsch, A. "Ueber die simultane Integration linearer partieller Differentialgleichungen", J. Refine. Angew. Math. (Crelle) 65 (1866) 257-268.
• Deahna, F. "Über die Bedingungen der Integrabilitat ....", J. Refine Angew. Math. 20 (1840) 340-350.
• Frobenius, G. "Über das Pfaffsche probleme", J. für Reine und Agnew. Math., 82 (1877) 230-315.