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On bivectors and jay-vectors

  • M. Hayes
  • N. H. ScottEmail author
Article
  • 18 Downloads

Abstract

A combination \(\varvec{a}+{\mathrm{i}}\varvec{b}\) where \({\mathrm{i}}^2=-1\) and \(\varvec{a},\,\varvec{b}\) 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 \(\varvec{a}+{\mathrm{j}}\varvec{b}\) of real vectors \(\varvec{a}\) and \(\varvec{b}\), where \({\mathrm{j}}^2=+1\) but \({\mathrm{j}}\) is not a real number, so \({\mathrm{j}}\ne \pm 1\). 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.

Keywords

Split complex numbers Hyperbolic numbers Coquaternions Conjugate semi-diameters Hyperboloids and ellipsoids PDEs 

Notes

References

  1. 1.
    Bell, R.J.T.: Coordinate Geometry of Three Dimensions. Macmillan, London (1937)Google Scholar
  2. 2.
    Boulanger, Ph, Hayes, M.: Bivectors and Waves in Mechanics and Optics. Chapman and Hall, London (1993)CrossRefzbMATHGoogle Scholar
  3. 3.
    Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Zampetti, P.: Geometry of Minkowski Space-Time, Chapter 2: Hyperbolic Numbers. Springer, Basel (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cockle, J.: On certain functions resembling quaternions and on a new imaginary in algebra. Philos. Mag. Ser. 3 33, 435–439 (1848)Google Scholar
  5. 5.
    Cockle, J.: On systems of algebra involving more than one imaginary; and on equations of the fifth degree. Philos. Mag. Ser. 3 35, 434–437 (1849)Google Scholar
  6. 6.
    Gantmacher, F.R.: Applications of the Theory of Matrices, vol. 2. Interscience Publishers, New York (1959)zbMATHGoogle Scholar
  7. 7.
    Gibbs, J. W.: Elements of Vector Analysis, 1881, 1884 (privately printed), OR pp. 17–90, Vol. 2, Part 2 Scientific Papers, Dover Publications, New York (1961)Google Scholar
  8. 8.
    Hamilton, W.R.: Lectures on Quaternions. Hodges and Smith, Dublin (1853)Google Scholar
  9. 9.
    Hayes, M.: Inhomogeneous plane waves. Arch. Rational Mech. Anal. 85, 41–79 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hayes, M.: Inhomogeneous electromagnetic plane waves in crystals. Arch. Rational Mech. Anal. 97, 221–260 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
  12. 12.
    Yaglom, I.M.: Complex Numbers in Geometry. Academic Press, London (1968)zbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.School of Mechanical and Materials EngineeringUniversity CollegeDublinIreland
  2. 2.School of MathematicsUniversity of East AngliaNorwichUK

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