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 right.

Introduction

One-form version

Suppose we are to find the trajectory of a particle in a subset of 3D space, but we do not know its trajectory formula. Instead, we know only that its trajectory satisfies ${\displaystyle adx+bdy+cdz=0}$, where ${\displaystyle a,b,c}$ are smooth functions of ${\displaystyle (x,y,z)}$. Thus, our only certainty is that if at some moment in time the particle is at location ${\displaystyle (x_{0},y_{0},z_{0})}$, then its velocity at that moment is restricted within the plane with equation

In other words, we can draw a "local plane" at each point in 3D space, and we know that the particle's trajectory must be tangent to the local plane at all times.

If we have two equations

then we can draw two local planes at each point, and their intersection is generically a line, allowing us to uniquely solve for the curve starting at any point. In other words, with two 1-forms, we can foliate the domain into curves.

If we have only one equation ${\displaystyle adx+bdy+cdz=0}$, then we might be able to foliate ${\displaystyle \mathbb {R} ^{3}}$ into surfaces, in which case, we can be sure that a curve starting at a certain surface must be restricted to wander within that surface. If not, then a curve starting at any point might end up at any other point in ${\displaystyle \mathbb {R} ^{3}}$.

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. If this happens, then we would not get a 2-dimensional surface, but a 3-dimensional blob. An example is shown in the diagram on the right.

If the one-form is integrable, then loops exactly close upon themselves, and each surface would be 2-dimensional. Frobenius' theorem states that this happens precisely when ${\displaystyle \omega \wedge d\omega =0}$ over all of the domain, where ${\displaystyle \omega :=adx+bdy+cdz}$. The notation is defined in the article on one-forms.

During his development of axiomatic thermodynamics, Carathéodory proved that if ${\displaystyle \omega }$ is an integrable one-form on an open subset of ${\displaystyle \mathbb {R} ^{n}}$, then ${\displaystyle \omega =fdg}$ for some scalar functions ${\displaystyle f,g}$ on the subset. This is usually called Carathéodory's theorem in axiomatic thermodynamics.^[1][2] One can prove this intuitively by first constructing the little planes according to ${\displaystyle \omega }$, quilting them together into a foliation, then assigning each surface in the foliation with a scalar label. Now for each point ${\displaystyle p}$, define ${\displaystyle g(p)}$ to be the scalar label of the surface containing point ${\displaystyle p}$.

Now, ${\displaystyle dg}$ is a one-form that has exactly the same planes as ${\displaystyle \omega }$. However, it has "even thickness" everywhere, while ${\displaystyle \omega }$ might have "uneven thickness". This can be fixed by a scalar scaling by ${\displaystyle f}$, giving ${\displaystyle \omega =fdg}$. This is illustrated on the right.

Multiple one-forms

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

${\displaystyle \left\{f_{k}^{i}:\mathbf {R} ^{n}\to \mathbf {R} \ :\ 1\leq i\leq n,1\leq k\leq r\right\}}$

be a collection of C¹ functions, with r < n, and such that the matrix ( fⁱ
_k ) 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:

${\displaystyle (1)\{\begin{array}{l}{L}_{1}u\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\underset{i}{\sum <!--\; \sum \; -->}{f}_1^i(x)\frac{\mathrm{\partial <!--\; \partial \; -->}u}{\mathrm{\partial <!--\; \partial \; -->}{x}^i}={\stackrel{\to <!--\; \to \; -->}{f}}_1\cdot <!--\; \cdot \; -->\mathrm{\nabla <!--\; \nabla \; -->}u\end{array}}$Zdroj:https://en.wikipedia.org?pojem=Frobenius_theorem_(differential_topology)
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