|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''Frobenius' theorem''' gives [[necessary and sufficient condition]]s for finding a maximal set of independent solutions of an [[underdetermined system]] of first-order homogeneous linear [[partial differential equation]]s. In modern [[differential geometry|geometric]] terms, the theorem gives necessary and sufficient conditions for the existence of a [[foliation]] by '''maximal''' [[integral manifold]]s each of whose tangent bundles are spanned by a given family of [[vector field]]s (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]].
| | Roberto is the name Method love to be with although it can not the name attached to my birth certificate. My [http://www.ehow.com/search.html?s=neighbours neighbours] say it's not really for me but the thing that I love doing should be to drive but Concerning been taking on unique things lately. South Carolina is where simple home is. I used to be [https://www.Flickr.com/search/?q=unemployed unemployed] but actually I am a cashier but the promotion absolutely not comes. I've been working on my website for some point in time now. Check it out of the house here: http://circuspartypanama.com<br><br> |
| ==Introduction==
| |
| <!--EDITORS: If you make changes to the index positions or their labels, please be sure to do so in a way that doesn't mess things up in other places! -->
| |
| 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 equation]]s. Suppose that ''f''<sub>''k''</sub><sup>i</sup>(''x'') are a collection of real-valued [[continuously differentiable|''C''<sup>1</sup>]] functions on ''R''<sup>''n''</sup>, for ''i'' = 1, 2, ..., ''n'', and ''k'' = 1, 2, ..., ''r'', where ''r'' < ''n'', such that the matrix (''f''<sub>''k''</sub><sup>''i''</sup>) has [[rank of a matrix|rank]] ''r''. Consider the following system of partial differential equations for a real-valued ''C''<sup>2</sup> function ''u'' on ''R''<sup>''n''</sup>:
| |
|
| |
|
| :<math>
| | Feel free to surf to my homepage ... [http://circuspartypanama.com clash of clans hack No survey] |
| \left.
| |
| \begin{matrix}
| |
| L_1u\ \stackrel{\mathrm{def}}{=}\ \sum_i f_1^i(x)\frac{\partial u}{\partial x^i} &= 0\\
| |
| L_2u\ \stackrel{\mathrm{def}}{=}\ \sum_i f_2^i(x)\frac{\partial u}{\partial x^i} &= 0\\
| |
| \dots&\\
| |
| L_ru\ \stackrel{\mathrm{def}}{=}\ \sum_i f_r^i(x)\frac{\partial u}{\partial x^i} &= 0
| |
| \end{matrix}\right\}
| |
| </math> (1)
| |
| | |
| One seeks conditions on the existence of a collection of solutions ''u''<sub>1</sub>, ..., ''u''<sub>''n''−''r''</sub> such that the gradients
| |
| | |
| :<math>\nabla u_1,\nabla u_2,\dots,\nabla u_{n-r}</math>
| |
| | |
| are [[linearly independent]].
| |
| | |
| The Frobenius theorem asserts that this problem admits a solution locally<ref>Here ''locally'' means inside small enough open subsets of '''R'''<sup>''n''</sup>. Henceforth, when we speak of a solution, we mean a local solution.</ref> if, and only if, the operators ''L''<sub>''k''</sub> satisfy a certain [[integrability condition]] known as ''involutivity''. Specifically, they must satisfy relations of the form
| |
| | |
| :<math>L_iL_ju(x)-L_jL_iu(x)=\sum_k c_{ij}^k(x)L_ku(x)</math>
| |
| | |
| for ''i'', ''j'' = 1, 2,..., ''r'', and all ''C''<sup>2</sup> functions ''u'', and for some coefficients ''c''<sup>''k''</sup><sub>''ij''</sub>(''x'') that are allowed to depend on ''x''.
| |
| In other words, the [[commutator]]s [''L''<sub>''i''</sub>,''L''<sub>''j''</sub>] must lie in the [[linear span]] of the ''L''<sub>''k''</sub> 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''<sub>''i''</sub> so that the resulting operators do commute, and then to show that there is a [[coordinate system]] ''y''<sub>''i''</sub> for which these are precisely the partial derivatives with respect to ''y''<sub>1</sub>, ..., ''y''<sub>''r''</sub>.
| |
| | |
| === From analysis to geometry ===
| |
| Solutions to underdetermined systems of equations are seldom unique. For example, the system
| |
| | |
| :<math>
| |
| \begin{matrix}\frac{\partial f}{\partial x}&+&\frac{\partial f}{\partial y}&&&=0\\
| |
| &&\frac{\partial f}{\partial y}&+&\frac{\partial f}{\partial z}&=0
| |
| \end{matrix}
| |
| </math>
| |
| | |
| 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 ''f''<sub>1</sub> and ''f''<sub>2</sub> are two different solutions, the [[level surface]]s of ''f''<sub>1</sub> and ''f''<sub>2</sub> must overlap. In fact, the level surfaces for this system are all planes in '''R'''<sup>3</sup> 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:
| |
| | |
| :<math>f(x,y,z)=C(t)\text{ whenever }x - y + z = t.\, </math>
| |
| | |
| 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 ''u''<sub>1</sub>,...,''u''<sub>''n''−''r''</sub> are solutions of the problem (1) satisfying the independence condition on the gradients. Consider the [[level set]]s<ref>A level set is a subset of ''R''<sup>n</sup> corresponding to the locus of:
| |
| | |
| :(''u''<sub>1</sub>,...,''u''<sub>n-r</sub>) = (''c''<sub>1</sub>,...,''c''<sub>''n''−''r''</sub>),
| |
| | |
| for some constants ''c''<sub>i</sub>.</ref> of (''u''<sub>1</sub>,...,''u''<sub>n-r</sub>) regarded as an ''R''<sup>''n''−''r''</sub>-valued function. If ''v''<sub>1</sub>,...,''v''<sub>''n''−''r''</sub> 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.<ref>The notion of a continuously differentiable function on a family of level sets can be made rigorous by means of the [[implicit function theorem]].</ref>
| |
| | |
| 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.
| |
| <!--Explain this better.-->
| |
| | |
| ==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 system]]s, which today can be translated into the language of [[differential form]]s. An alternative formulation, which is somewhat more intuitive, uses [[vector field]]s.
| |
| | |
| ===Formulation using vector fields===
| |
| 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]]. In this context, the Frobenius theorem relates [[Integrability conditions for differential systems|integrability]] to foliation; to state the theorem, both concepts must be clearly defined.
| |
| | |
| One begins by noting that an arbitrary smooth [[vector field]] ''X'' on a manifold ''M'' can be integrated to define a family of [[curve]]s. 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.
| |
| | |
| This idea of integrability can be extended to collections of vector fields as well. One says that a [[subbundle]] <math>E\subset TM</math> of the [[tangent bundle]] ''TM'' is '''integrable''' (or '''involutive'''), if, for any two vector fields ''X'' and ''Y'' taking values in ''E'', then the [[Lie bracket of vector fields|Lie bracket]] <math>[X,Y]</math> 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''.
| |
| | |
| A subbundle <math>E\subset TM</math> may also be defined to arise from a [[foliation]] of a manifold. Let <math>N\subset M</math> be a submanifold that is a leaf of a foliation. Consider the tangent bundle ''TN''. If ''TN'' is exactly ''E'' with base space 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 [[chart (topology)|charts]].
| |
| | |
| 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'', ''Ω<sup>1</sup>(U)'' be the space of smooth, differentiable [[1-form]]s on ''U'', and ''F'' be a [[submodule]] of ''Ω<sup>1</sup>(U)'' of [[rank (linear algebra)|rank]] ''r'', the rank being constant in value over ''U''. The Frobenius theorem states that ''F'' is [[Integrability conditions for differential systems|integrable]] if and only if for every <math>p\in U</math> the [[stalk of a sheaf|stalk]] ''F<sub>p</sub>'' is generated by ''r'' [[exact differential form]]s.
| |
| | |
| 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 form]]s and [[Lie derivative]]s. Frobenius' theorem is one of the basic tools for the study of [[vector field]]s and foliations.
| |
| | |
| There are thus two forms of the theorem: one which operates with [[Distribution_(differential_geometry)|distribution]]s, that is smooth subbundles ''D'' of the tangent bundle T''M''; and the other which operates with subbundles of the graded ring Ω(''M'') of all forms on ''M''. These two forms are related by duality. If ''D'' is a smooth tangent distribution on ''M'', then the annihilator of ''D'', ''I''(''D'') consists of all forms α ∈ Ω(''M'') such that
| |
| | |
| :<math>i_v\alpha = 0\,</math>
| |
| | |
| for all ''v'' ∈ ''D'', where ''i'' denotes the [[interior product]] of a vector field with a ''k''-form. The set ''I''(''D'') forms a subring and, in fact, an ideal in ''Ω(M)''. Furthermore, using the definition of the [[exterior derivative]], it can be shown that ''I''(''D'') is closed under exterior differentiation (it is a [[differential ideal]]) if and only if ''D'' is involutive. Consequently, the Frobenius theorem takes on the equivalent form that ''I''(''D'') is closed under exterior differentiation if and only if ''D'' is integrable.
| |
| | |
| ==Generalizations==
| |
| The theorem may be generalized in a variety of ways.
| |
| | |
| ===Infinite dimensions===
| |
| One infinite-dimensional generalization is as follows.<ref>{{cite book|author=Dieudonné, J|title=Foundations of modern analysis|publisher=Academic Press|year=1969|pages=Chapter 10.9|nopp=true}}</ref> Let ''X'' and ''Y'' be [[Banach space]]s, and ''A'' ⊂ ''X'', ''B'' ⊂ ''Y'' a pair of [[open set]]s. Let
| |
| | |
| :<math>F:A\times B \to L(X,Y)</math>
| |
| | |
| be a [[continuously differentiable function]] of the [[Cartesian product]] (which inherits a [[differentiable structure]] from its inclusion into ''X''×''Y'') into the space ''L''(''X'',''Y'') of [[continuous linear transformation]]s of ''X'' into ''Y''. A differentiable mapping ''u'' : ''A'' → ''B'' is a solution of the differential equation
| |
| | |
| :<math>y' = F(x,y) </math> (1)
| |
| | |
| if ''u''′(''x'') = ''F''(''x'',''u''(''x'')) for all ''x'' ∈ ''A''.
| |
| | |
| The equation (1) is '''completely integrable''' if for each <math>(x_0, y_0)\in A\times B</math>, there is a neighborhood ''U'' of ''x''<sub>0</sub> such that (1) has a unique solution ''u''(''x'') defined on ''U'' such that ''u''(''x''<sub>0</sub>)=''y''<sub>0</sub>.
| |
| | |
| The conditions of the Frobenius theorem depend on whether the underlying [[field (mathematics)|field]] is '''R''' or '''C'''. If it is '''R''', then assume ''F'' is continuously differentiable. If it is '''C''', then assume ''F'' is twice continuously differentiable. Then (1) is completely integrable at each point of ''A''×''B'' if and only if
| |
| | |
| :<math>D_1F(x,y)\cdot(s_1,s_2) + D_2F(x,y)\cdot(F(x,y)\cdot s_1,s_2)</math>
| |
| :::<math>=D_1F(x,y)\cdot (s_2,s_1) + D_2F(x,y)\cdot(F(x,y)\cdot s_2,s_1)</math>
| |
| for all ''s''<sub>1</sub>, ''s''<sub>2</sub> ∈ ''X''. Here ''D''<sub>1</sub> (resp. ''D''<sub>2</sub>) denotes the partial derivative with respect to the first (resp. second) variable; the dot product denotes the action of the linear operator ''F''(''x'',''y'') ∈ ''L''(''X'',''Y''), as well as the actions of the operators ''D''<sub>1</sub>''F''(''x'',''y'') ∈ ''L''(''X'',''L''(''X'',''Y'')) and ''D''<sub>2</sub>''F''(''x'',''y'') ∈ ''L''(''Y'', ''L''(''X'',''Y'')).
| |
| | |
| ==== Banach manifolds ====
| |
| The infinite dimensional version of the Frobenius theorem also holds on [[Banach manifold]]s.<ref>{{cite book|author=Lang, S.|title=Differential and Riemannian manifolds|publisher=Springer-Verlag|year=1995|isbn=978-0-387-94338-1|pages=Chapter VI: The theorem of Frobenius|nopp=true}}</ref> The statement is essentially the same as the finite dimensional version.
| |
| | |
| Let ''M'' be a Banach manifold of class at least ''C''<sup>2</sup>. 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 ''E''<sub>p</sup>:
| |
| | |
| :<math> [X,Y]_p \in E_p</math>
| |
| | |
| 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 [[pushforward (differential)|differential]] of φ is an isomorphism of ''TN'' with φ<sup>-1</sup>''E''.
| |
| | |
| The Frobenius theorem states that a subbundle ''E'' is integrable if and only if it is involutive.
| |
| | |
| ===Holomorphic forms===
| |
| The statement of the theorem remains true for [[complex differential form#Holomorphic forms|holomorphic 1-forms]] on [[complex manifold]]s — manifolds over '''C''' with biholomorphic [[transition function]]s.<ref>{{cite book | author=Kobayashi, Shoshichi and Nomizu, Katsumi | title = [[Foundations of Differential Geometry]], Vol. 2 | publisher=[[Wiley Interscience]] | year=1969|pages=Appendix 8 | nopp=true}}</ref>
| |
| | |
| Specifically, if <math>\omega^1,\dots,\omega^r</math> are ''r'' linearly independent holomorphic 1-forms on an open set in '''C'''<sup>n</sup> such that
| |
| | |
| :<math>d\omega^j = \sum_{i=1}^r \psi_i^j \wedge \omega^i</math>
| |
| | |
| for some system of holomorphic 1-forms ψ<sub>i</sub><sup>j</sup>, ''i'',''j''=1,...,r, then there exist holomorphic functions ''f''<sub>i</sub><sup>j</sup> and ''g''<sup>i</sup> such that, on a possibly smaller domain,
| |
| | |
| :<math>\omega^j = \sum_{i=1}^r f_i^jdg^i.</math>
| |
| | |
| This result holds locally in the same sense as the other versions of the Frobenius theorem. In particular, the fact that it has been stated for domains in '''C'''<sup>n</sup> is not restrictive.
| |
| | |
| ===Higher degree forms===
| |
| 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]].
| |
| | |
| ==History==
| |
| Despite being named for [[Ferdinand Georg Frobenius]], the theorem was first proven by [[Alfred Clebsch]] and [[Feodor 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 system]]s, thus paving the way for its usage in differential topology.
| |
| | |
| ==See also==
| |
| * [[Integrability conditions for differential systems]]
| |
| * [[Domain-straightening theorem]]
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==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. Reine. Angew. Math. (Crelle)'' '''65''' (1866) 257-268.
| |
| * Deahna, F. "Über die Bedingungen der Integrabilitat ....", ''J. Reine Angew. Math.'' '''20''' (1840) 340-350.
| |
| * Frobenius, G. "Über das Pfaffsche probleme", ''J. für Reine und Agnew. Math.'', '''82''' (1877) 230-315.
| |
| | |
| [[Category:Theorems in differential geometry]]
| |
| [[Category:Theorems in differential topology]]
| |
| [[Category:Differential systems]]
| |
| [[Category:Foliations]]
| |