Poincaré-Bohl's theorem

The set of Poincaré Bohl , English Poincaré Bohl theorem is a theorem from the mathematical sub-region of the topology of which the two mathematicians Henri Poincaré and Piers Bohl is allocated. The theorem represents a fundamental property of Brouwer's degree of mapping for continuous vector fields in real coordinate space . This property is also called linear homotopy and results directly from the homotopy invariance of the degree of mapping.

Formulation of the sentence

According to the presentation by Alexandroff - Hopf and Ortega - Rheinboldt , the sentence can be stated as follows:

Let an open and bounded set and two continuous mappings be given ${\displaystyle \Omega \subset \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$

${\displaystyle F,G\colon {\overline {\Omega }}\to \mathbb {R} ^{n}}$  .

To this end

${\displaystyle \gamma =\partial {\Omega }}$

the associated set of edge points as well as

${\displaystyle U:=\bigcup _{x\in \gamma }[F(x)\;,\;G(x)]}$

the set of all points that lie on the connecting lines between the - and - image points of the respective edge points. ${\displaystyle F}$ ${\displaystyle G}$

Then:

For every point lying outside, i.e. for every point , the Brouwer's degrees of mapping of and agree: ${\displaystyle y\in \mathbb {R} ^{n}\setminus U}$ ${\displaystyle F}$ ${\displaystyle G}$

${\displaystyle d(F,\Omega ,y)=d(G,\Omega ,y)}$  .

literature

• P. Alexandroff, H. Hopf: Topologie (=  The basic teachings of the mathematical sciences . Volume 45 ). First volume (reprint). Chelsea Publishing Company, New York 1965 ( MR0185557 ).
• P. Bohl: About the movement of a mechanical system near a position of equilibrium . In: Journal for pure and applied mathematics . tape 127 , 1904, pp. 179-276 ( MR1580639 ).
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• James M. Ortega, WC Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables . (Unabridged republication of the work first published by Academic Press, New York and London, 1970) (=  Classics in Applied Mathematics . Volume 30 ). Society for Industrial and Applied Mathematics, Philadelphia 2000, ISBN 0-89871-461-3 ( MR1744713 ).
• H. Poincaré: Sur les courbes définies par les equations différentielles IV . In: Journal de Mathématiques Pures et Appliquées . tape 2 , 1886, p. 151-217 .
• Eberhard Zeidler : Nonlinear Functional Analysis and its Applications . I. Fixed-point theorems. Springer Verlag, New York ( inter alia) 1986 ( MR0816732 ).

References and footnotes

1. ↑ ^{a b} P. Alexandroff, H. Hopf: Topology I . 1965, p. 459
2. ↑ ^{a b} J. M. Ortega, WC Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables 2000, p. 157
3. ↑ Eberhard Zeidler: Nonlinear Functional Analysis and its Applications I . 1986, p. 570 ff
4. ↑ From the presentation in the introduction to the combinatorial topology by Egbert Harzheim it can be seen that from the Poincaré-Bohl theorem, even the famous Poincaré-Brouwer theorem can be deduced directly through elementary inferences ( b: evidence archive: topology: the sentence von Poincaré-Bohl implies the Poincaré-Brouwer theorem ).
5. ↑ is the closed envelope of  . ${\displaystyle {\overline {\Omega }}}$ ${\displaystyle \Omega }$