TY - BOOK

T1 - A complex-like calculus for spherical vectorfields

AU - Graaf, de, J.

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

M3 - Report

T3 - CASA-report

BT - A complex-like calculus for spherical vectorfields

PB - Technische Universiteit Eindhoven

CY - Eindhoven

ER -