The curl in seven dimensional space and its applications
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.
KeywordsCross Product Maxwell Equation Division Algebra Natural Concept Hadamard Matrix
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- Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, GTM 82, Springer-Verlag, 1982.Google Scholar
- Hodge, W. W. D., The Theory and Applications of Harmonic Integrals, Cambridge, Univ. Press, 1941.Google Scholar
- Jacobson, N. Basic Algebra, Vol. 1, W. H. Freeman and Company, 1974.Google Scholar
- Salamon, D., Spin Geometry and Seiberg-Witten Invariants, Preprint, 1997.Google Scholar
- Wandi Wei, Combinatorial Theory (second volume), Scientific Publishing House (in Chinese), 1987.Google Scholar
- Whitehead, G. W., Elements of Homotopy Theory, GTM 61, Springer-Verlag, 1978.Google Scholar
- Yosida, K., Functional Analysis, Sixth Edition. Springer-Verlag, 1980.Google Scholar