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Differential forms and Clifford analysis

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Abstract
In this paper we use a calculus of differential forms which is defined using an axiomatic approach. We then define integration of differential forms over chains in a new way and we present a short proof of Stokes' formula using distributional techniques. We also consider differential forms in Clifford analysis, vector differentials and their powers. This framework enables an easy proof for a Cauchy formula on a k-surface. Finally, we discuss how to compute winding numbers in terms of the monogenic Cauchy kernel and the vector differentials with a new approach which does not involve cohomology of differential forms.
Keywords
Differential forms, Clifford algebras, monogenic functions, winding numbers

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MLA
Sabadini, Irene, and Franciscus Sommen. “Differential Forms and Clifford Analysis.” Modern Trends in Hypercomplex Analysis, edited by Swanhild Bernstein et al., Birkhäuser, 2016, pp. 247–64, doi:10.1007/978-3-319-42529-0_14.
APA
Sabadini, I., & Sommen, F. (2016). Differential forms and Clifford analysis. In S. Bernstein, U. Kähler, I. Sabadini, & F. Sommen (Eds.), Modern trends in hypercomplex analysis (pp. 247–264). https://doi.org/10.1007/978-3-319-42529-0_14
Chicago author-date
Sabadini, Irene, and Franciscus Sommen. 2016. “Differential Forms and Clifford Analysis.” In Modern Trends in Hypercomplex Analysis, edited by Swanhild Bernstein, Uwe Kähler, Irene Sabadini, and Franciscus Sommen, 247–64. Cham: Birkhäuser. https://doi.org/10.1007/978-3-319-42529-0_14.
Chicago author-date (all authors)
Sabadini, Irene, and Franciscus Sommen. 2016. “Differential Forms and Clifford Analysis.” In Modern Trends in Hypercomplex Analysis, ed by. Swanhild Bernstein, Uwe Kähler, Irene Sabadini, and Franciscus Sommen, 247–264. Cham: Birkhäuser. doi:10.1007/978-3-319-42529-0_14.
Vancouver
1.
Sabadini I, Sommen F. Differential forms and Clifford analysis. In: Bernstein S, Kähler U, Sabadini I, Sommen F, editors. Modern trends in hypercomplex analysis. Cham: Birkhäuser; 2016. p. 247–64.
IEEE
[1]
I. Sabadini and F. Sommen, “Differential forms and Clifford analysis,” in Modern trends in hypercomplex analysis, Macau, PEOPLES R CHINA, 2016, pp. 247–264.
@inproceedings{8502201,
  abstract     = {{In this paper we use a calculus of differential forms which is defined using an axiomatic approach. We then define integration of differential forms over chains in a new way and we present a short proof of Stokes' formula using distributional techniques. We also consider differential forms in Clifford analysis, vector differentials and their powers. This framework enables an easy proof for a Cauchy formula on a k-surface. Finally, we discuss how to compute winding numbers in terms of the monogenic Cauchy kernel and the vector differentials with a new approach which does not involve cohomology of differential forms.}},
  author       = {{Sabadini, Irene and Sommen, Franciscus}},
  booktitle    = {{Modern trends in hypercomplex analysis}},
  editor       = {{Bernstein, Swanhild and Kähler, Uwe and Sabadini, Irene and Sommen, Franciscus}},
  isbn         = {{9783319425283}},
  issn         = {{2297-0215}},
  keywords     = {{Differential forms,Clifford algebras,monogenic functions,winding numbers}},
  language     = {{eng}},
  location     = {{Macau, PEOPLES R CHINA}},
  pages        = {{247--264}},
  publisher    = {{Birkhäuser}},
  title        = {{Differential forms and Clifford analysis}},
  url          = {{http://doi.org/10.1007/978-3-319-42529-0_14}},
  year         = {{2016}},
}

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