# Frobenius Theorem (differential Topology) - Introduction

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 fki(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 (fki) has rank r. Consider the following system of partial differential equations for a real-valued C2 function u on Rn:

$left. begin{matrix} L_1u stackrel{mathrm{def}}{=} sum_i_f_1^i(x)frac{partial u}{partial x^i} &= 0\ L_2u stackrel{mathrm{def}}{=} sum_i_f_2^i(x)frac{partial u}{partial x^i} &= 0\ dots&\ L_ru stackrel{mathrm{def}}{=} sum_i_f_r^i(x)frac{partial u}{partial x^i} &= 0 end{matrix}right}.$ (1)

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

are linearly independent.

The Frobenius theorem asserts that this problem admits a solution locally if, and only if, the operators Lk satisfy a certain integrability condition known as involutivity. Specifically, they must satisfy relations of the form

for i, j = 1, 2,..., r, and all C2 functions u, and for some coefficients ckij(x) that are allowed to depend on x. In other words, the commutators must lie in the linear span of the Lk 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 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.

### Other articles related to "introduction":

China Miéville - Bibliography - Nonfiction - Introductions To Fiction By Other Authors
... The Borribles An Introduction, 2001 ... Things That Never Happen An Introduction, 2002 ... Wizardry and Wild An Introduction, 2004 ...
John Frame (theologian) - Selected Works
... Introduction to Presuppositional Apologetics Part 2 ... Van Til The Theologian, 1976 ISBN 0-916034-02-X Medical Ethics, 1988 ISBN 0-87552-261-0 ...
Introduction - Music - Songs and Tracks
... Introduction", by Chicago from The Chicago Transit Authority "Introduction", by Hood from Outside Closer "Introduction", by Kajagoogoo from White Feathers "Introduction", by Mike ...
Introduction To Metaphysics
... An Introduction to Metaphysics (Introduction à la Métaphysique) is a 1903 essay by Henri Bergson that explores the concept of reality ...

### Famous quotes containing the word introduction:

Such is oftenest the young man’s introduction to the forest, and the most original part of himself. He goes thither at first as a hunter and fisher, until at last, if he has the seeds of a better life in him, he distinguishes his proper objects, as a poet or naturalist it may be, and leaves the gun and fish-pole behind. The mass of men are still and always young in this respect.
Henry David Thoreau (1817–1862)

For the introduction of a new kind of music must be shunned as imperiling the whole state; since styles of music are never disturbed without affecting the most important political institutions.
Plato (c. 427–347 B.C.)

We used chamber-pots a good deal.... My mother ... loved to repeat: “When did the queen reign over China?” This whimsical and harmless scatological pun was my first introduction to the wonderful world of verbal transformations, and also a first perception that a joke need not be funny to give pleasure.
Angela Carter (1940–1992)