On bivectors and jay-vectors

Hayes, M. and Scott, N. H. (2019) On bivectors and jay-vectors. Ricerche di Matematica, 68 (2). pp. 859-882. ISSN 0035-5038

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Abstract

A combination a+ib where i2=−1 and a, b are real vectors is called a bivector. Gibbs developed a theory of bivectors, in which he associated an ellipse with each bivector. He obtained results relating pairs of conjugate semi-diameters and in particular considered the implications of the scalar product of two bivectors being zero. This paper is an attempt to develop a similar formulation for hyperbolas by the use of jay-vectors—a jay-vector is a linear combination a+jb of real vectors a and b, where j2=+1 but j is not a real number, so j≠±1. The implications of the vanishing of the scalar product of two jay-vectors is also considered. We show how to generate a triple of conjugate semi-diameters of an ellipsoid from any orthonormal triad. We also see how to generate in a similar manner a triple of conjugate semi-diameters of a hyperboloid and its conjugate hyperboloid. The role of complex rotations is discussed briefly. Application is made to second order elliptic and hyperbolic partial differential equations.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: LivePure Connector
Date Deposited: 15 Apr 2019 11:30
Last Modified: 29 Jun 2020 23:57
URI: https://ueaeprints.uea.ac.uk/id/eprint/70573
DOI: 10.1007/s11587-019-00442-2

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