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In this paper we express the theory of harmonic differential forms on conformal Euclidean manifolds in terms of the so called Clifford multivector fields. The aim is to give good definitions for d and d* operators in Clifford multivector case. Using these definitions we derive a formula for the Laplace operator. Three fundamental examples are included in the end of the paper and connections to existing theory is discussed.

Let Fr (0 < r < m + 1) be a smooth r-vector valued function in a suitable open domain of \({\mathbb{R}}^{m+1}\) satisfying \(\partial F_r = 0\) in Ω, where ∂ is the Dirac operator in \({\mathbb{R}}^{m+1}\) . Then it is proved that there exists Hr, an r-vector valued harmonic function in Ω, such that Fr = \(\partial H_r\partial\). Two proofs of this structure theorem are given, one based on properties of harmonic differential forms and one relying upon primitivation of monogenic functions.

The space HFk(Ω) of harmonic multi-vector fields in a domain \(\Omega \subset \mathbb{R}^{n}\) as introduced in [1] is closely connected to the space of harmonic forms. The main aim of this paper is to characterize the dual space of HFk(E) being \(\mathbf{E} \subset \mathbb{R}^{n}\) a compact set. It is proved that HFk(E)* is isomorphic to a certain quotient space of so-called harmonic pairs outside E vanishing at infinity.