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A Unified Constructive Approach to the Topological Degree in R n
Author(s) -
Kliesch Wolfgang
Publication year - 1989
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19891420113
Subject(s) - degree (music) , polyhedron , mathematics , bounded function , function (biology) , computation , boundary (topology) , regular polygon , constructive , simple (philosophy) , combinatorics , topology (electrical circuits) , discrete mathematics , mathematical analysis , geometry , algorithm , computer science , philosophy , physics , acoustics , biology , operating system , process (computing) , epistemology , evolutionary biology
The paper gives an approach to the topological degree in R n which takes into account numerical requirements and permits derivation of the known degree computation formulas in a simple way. The new approach subsumes several earlier approaches and represents a general principle of construction of degree computation formulas. The basic idea consists of computing the degree of a continuous function relative to a bounded open subset Ω of R n by means of an auxiliary function which is defined on a polyhedron approximating Ω and maps into a known fixed convex polyhedron containing the origin of R n . It is further shown that the topological degree of a continuous function relative to an n ‐dimensional polyhedron P can be computed alone by means of a subset of the boundary of P .