A complex-like calculus for spherical vectorfields

J. Graaf, de

Onderzoeksoutput: Boek/rapportRapportAcademic

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First, R^{1+d}, d in N, is turned into an algebra by mimicing the usual complex multiplication. Indeed the special case d = 1 reproduces C. For d > 1 the considered algebra is commutative, but non-associative and even non-alternative. Next, the Dijkhuis class of mappings (’vectorfields’) R^{1+d} ¿ R^{1+d}, suggested by C.G. Dijkhuis for d=3, d=7, is introduced. This special class is then fully characterized in terms of analytic functions of one complex variable. Finally, this characterization enables to show easily that the Dijkhuis-class is closed under pointwise R^{d+1}-multiplication: It is a commutative and associative algebra of vector fields. Previously it had not been observed that the Dijkhuis-class only contains vectorfields with a ’time-dependent’ spherical symmetry. Such disappointment was to be expected! The class of functions which are differentiable with respect to the algebraic structure, that we impose on R^{1+d}, contains only linear functions if d > 1. The Dijkhuis-class does not appear this way either! In our treatment neither quaternions nor octonions play a role.
Originele taal-2Engels
Plaats van productieEindhoven
UitgeverijTechnische Universiteit Eindhoven
Aantal pagina's7
StatusGepubliceerd - 2011

Publicatie series

ISSN van geprinte versie0926-4507

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