Approximation Theory and its Applications

, Volume 15, Issue 3, pp 66-80

First online:

The curl in seven dimensional space and its applications

  • Lizhong PengAffiliated withDepartment of Mathematics, Peking University
  • , Lei YangAffiliated withDepartment of Mathematics, Peking University

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In higher dimensional spacesR n(n>3) the usual curl does not have the properties as inR 3. In this paper, we established the natural concept of curl inR 7 via octonion O. We prove that there exists the curl inR n if and only if n=3,7. Some applications are presented, such as the new phenomenon of the differential forms inR 7 which is different from the ordinary de Rham cohomology and Hodge theory, grad-curl-div type Dirac operator inR 6, seven dimensional Maxwell equations and Navier-Stokes equations.