Approximation Theory and its Applications

, Volume 15, Issue 3, pp 66–80

The curl in seven dimensional space and its applications

  • Lizhong Peng
  • Lei Yang


In higher dimensional spacesRn(n>3) the usual curl does not have the properties as inR3. In this paper, we established the natural concept of curl inR7 via octonion O. We prove that there exists the curl inRn if and only if n=3,7. Some applications are presented, such as the new phenomenon of the differential forms inR7 which is different from the ordinary de Rham cohomology and Hodge theory, grad-curl-div type Dirac operator inR6, seven dimensional Maxwell equations and Navier-Stokes equations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Atiyah, M. F. and Singer, I. M., The Index of Elliptic Operators on Compact Manifolds. Bull. Am. Math. Soc., 69 (1963), 422–433.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, GTM 82, Springer-Verlag, 1982.Google Scholar
  3. [3]
    Brown, R. B. and Gray, A., Vector Cross Products Comment, Math. Helv. 42 (1967), 222–236.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Eckmann, B., Stetige Lösungen Linearer Gleichungssysteme. Comment. Math. Helv., 15 (1942–3), 318–339.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Hodge, W. W. D., The Theory and Applications of Harmonic Integrals, Cambridge, Univ. Press, 1941.Google Scholar
  6. [6]
    Jacobson, N. Basic Algebra, Vol. 1, W. H. Freeman and Company, 1974.Google Scholar
  7. [7]
    Salamon, D., Spin Geometry and Seiberg-Witten Invariants, Preprint, 1997.Google Scholar
  8. [8]
    Wandi Wei, Combinatorial Theory (second volume), Scientific Publishing House (in Chinese), 1987.Google Scholar
  9. [9]
    Whitehead, G. W., Elements of Homotopy Theory, GTM 61, Springer-Verlag, 1978.Google Scholar
  10. [10]
    Yosida, K., Functional Analysis, Sixth Edition. Springer-Verlag, 1980.Google Scholar

Copyright information

© Springer 1999

Authors and Affiliations

  • Lizhong Peng
    • 1
  • Lei Yang
    • 1
  1. 1.Department of MathematicsPeking UniversityBeijingPRC

Personalised recommendations