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Differential Forms in Geometric Calculus

  • David Hestenes
Conference paper
Part of the Fundamental Theories of Physics book series (FTPH, volume 55)

Abstract

Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system of Geometric Calculus. The most significant application of the system is to relativistic physics where it is referred to as Spacetime Calculus. The fundamental integral theorems are discussed along with applications to physics, especially electrodynamics.

Key words

Differential forms geometric calculus Clifford algebra Dirac operator Stokes’ Theorem manifolds. 

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References

  1. 1.
    Hestenes, D.: 1986, ‘A Unified Language for Mathematics and Physics,’ Clifford Algebras and their Applications in Mathematical Physics, J.S.R. Chisholm/A.K. Common (eds.), Reidel, Dordrecht/Boston, pp. 1–23.Google Scholar
  2. 2.
    Hestenes, D.: 1988, ‘Universal Geometric Algebra,’ Simon Stevin 82, pp. 253–274.MathSciNetGoogle Scholar
  3. 3.
    Hestenes, D.: 1991, ‘Mathematical Viruses,’ Clifford Algebras and their Applications in Mathematical Physics, A. Micali, R. Boudet, J. Helmstetter (eds.), Kluwer, Dordrecht/Boston, pp. 3–16.Google Scholar
  4. 4.
    Hestenes, D.: 1993, ‘Hamiltonian Mechanics with Geometric Calculus,’ Z. Oziewicz, A. Borowiec, B. Jancewicz (eds.), Spinors, Twistors and Clifford Algebras Kluwer, Dordrecht/Boston.Google Scholar
  5. 5.
    Hestenes, D. and Sobczyk, G.: 1984, CLIFFORD ALGEBRA TO GEOMETRIC CALCULUS, A Unified Language fo Mathematics and Physics, D. Reidel Publ. Co., Dordrecht, paperback 1985, Third printing 1992.Google Scholar
  6. 6.
    Doran, C., Hestenes, D., Sommen, F. & Van Acker, N.: `Lie Groups as Spin Groups,’ Journal of Mathematical Physics (accepted).Google Scholar
  7. 7.
    O’Neill, B.: 1983, Semi-Riemannian Geometry, Academic Press, London.zbMATHGoogle Scholar
  8. 8.
    Hestenes, D.: 1966, Space-Time Algebra, Gordon and Breach, New York.zbMATHGoogle Scholar
  9. 9.
    Wills, A.P.: 1958, Vector Analysis with an Introduction to Tensor Analysis, Dover, New York.zbMATHGoogle Scholar
  10. 10.
    Hestenes, D.: 1968, `Multivector Calculus,’ J. Math. Anal. and Appl 24, pp. 313–325.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hestenes, D.: 1968, ‘Multivector Functions,’ J. Math. Anal. and Appl 24, pp. 467–473.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Delanghe, R.: 1970, ‘On regular-analytic functions with values in a Clifford algebra,’ Math. Ann 185, pp. 91–111.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sobczyk, G.: 1992, `Simplicial Calculus with Geometric Algebra,’ Clifford Algebras and Their Applications in Mathematical Physics, A. Micali, R. Boudet and J. Helmstetter (eds.), Kluwer, Dordrecht/Boston, pp. 279–292.Google Scholar
  14. 14.
    Hestenes, D.: 1974, `Proper Particle Mechanics,’ J. Math. Phys 15, 1768–1777.ADSCrossRefGoogle Scholar
  15. 15.
    Barut, A.: 1980, Electrodynamics and the classical theory of fields and particles, Dover, New York.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • David Hestenes
    • 1
  1. 1.Department of Physics and AstronomyArizona State University TempeUSA

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