Differential Forms and Clifford Analysis
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.
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Differential Forms and Clifford Analysis Irene Sabadini and Franciscus Sommen 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. Mathematics Subject Classification (2010). MSC: 58A10, 55M25, 30G35. Keywords. Differential forms, Clifford algebras, monogenic functions, winding numbers.
1. Introduction This paper is a continuation of our former papers [9, 10, 11, 12] in which the calculus of differential forms has been combined with the Clifford algebra. Using Clifford analysis techniques, and monogenic functions in particular, we were able to establish a Cauchy-type formula for the Dirac operator on surfaces (see [10]), a theory of monogenic differential forms allowing a cohomology theory (see [9, 12]) and a formula for the winding number of a k-cycle and a (m − k − 1)-cycle in Rm (see [9]). This extends the work of Hodge [7] in which the homology of a domain is measured in terms of integrals over cycles of harmonic differential forms. To understand these ideas, one has to recall that the theory of monogenic functions in Clifford analysis deals with nullsolutions of the Dirac operator ∂x in Rm , which is a higher-dimensional generalization of the theory of holomorphic functions in the plane. Consider a point p in the plane (or a number of points) and a closed Jordan curve (a 1-cycle) Γ ⊂ C \ {p}; then the winding number of Γ around p is given by the Cauchy integral dz 1 2πi Γ z − p
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which is a special case of the residue formula 1 f (z) dz . 2πi Γ z − p The analog Cauchy formula for monogenic functions has the form (see [1]) f (x) = E(u − x) σ(du) f (u) ∂C
where C is an open bounded set in Rm , x ∈ C, E(u − x) is the Cauchy kernel and σ(du) is a suitable (m − 1)-form with values in a Clifford algebra that represents the oriented surface measure. Using this Cauchy formula in special cases, one can establish a formula for the winding number of an (m − 1)-cycle around one or several points. However, in Rm one can also consider k-cycles Ck and (m − k − 1)-cycles Cm−k−1 in Rm \ Ck for which there is a winding number that can be defined in terms of the intersection number; it cannot be measured in terms of monogenic functions right away. This makes it necessary to combine a calculus of differential forms with the theory of monogenic functions, as we do in this work. The paper consists of 5 sections, besides this introduction. In Section 2, we define the calculus of differential forms from scratch using an axiomatic approach which is inspi
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