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

Wolfgang Kliesch

doi:10.4171/zaa/112

JournalszaaVol. 3, No. 4pp. 337–355

Zur numerischen Bestimmung des Abbildungsgrades im $R^{n}$ I

Wolfgang Kliesch
Universität Leipzig, Germany

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

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Abstract

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

deg (Φ, int P^{n}) = k^{- 1} σ \in T_{n - 1} \sum sgn d (σ, Φ),

d (σ, Φ := det (f (a^{1}) \dots f (a^{n})), σ = [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 $n$ -dimensional unit octahedron.

Cite this article

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

DOI 10.4171/ZAA/112