Zur numerischen Bestimmung des Abbildungsgrades im Rn\mathbb R^n I

  • Wolfgang Kliesch

    Universität Leipzig, Germany

Abstract

Two formulas computing the topological degree of a continuous function Φ\Phi relative to an nn-dimensional polyhedron PnP^n are presented. Both formulas are based on the same idea of construction. Let T be a triangulation of the boundary of PnP^n and let f(c)(c) = sgn Φ(e),e\Phi (e), e \in E(T), be a simplicial mapping from T into a boundary triangulation of the nn-dimensional unit cube, then the topological degree of the function Φ\Phi relative to PnP^n is given by

deg(Φ,int  Pn)=k1σTn1sgn  d(σ,Φ),\mathrm {deg} (\Phi, \mathrm {int} \; P^n) = k^{-1} \sum_{\sigma \in \mathrm T_{n-1}} \mathrm {sgn \; d}(\sigma, \Phi),
d(σ,Φ:=det(f(a1)f(an)),σ=[a1an],\mathrm d(\sigma, \Phi := \mathrm {det} (\mathrm f (a^1) \dots \mathrm f (a^n)), \quad \sigma = [a^1 \dots a^n],

if T is oriented in a suitable manner.

The second computation formula is based on a simplicial mapping from T into the natural boundary triangulation of the nn-dimensional unit octahedron.

Cite this article

Wolfgang Kliesch, Zur numerischen Bestimmung des Abbildungsgrades im Rn\mathbb R^n I. Z. Anal. Anwend. 3 (1984), no. 4, pp. 337–355

DOI 10.4171/ZAA/112