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Extra info for 1973, Year of the Humanoids: An Analysis of the Fall UFO Humanoid Wave

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See [78, Chapter VIII]. We will prove the duality by viewing the flat chains as currents. 9. Flat chains as currents. The theory of currents is an extension of the theory of distributions; currents act on forms of any given degree. More precisely, denote by Dk (Rn ) the vector space of smooth compactly supported k-forms in Rn (the test forms). 21) |T (ω)| ≤ C max ||∂ α ω||∞ |α|≤N for every ω ∈ Dk (Rn ) with support in K. 21) is taken over all partial derivatives of the components of ω up to order N .

Polyhedral chains are rather discrete objects; they can be thought of as finite valued functions supported on simplexes, with orientation regarded. Next we describe another Banach space, whose members are more diffused chains. This space ultimately will be shown to agree with the space Fk (Rn ) of flat k-chains. 6). 34) Rn dω, ψ dx Rn for every smooth compactly supported k-form ω. It is easy to see that such a vector field div ψ, if exists, is unique. 34) holds in the following two more general instances: ω is a flat form of compact support, or ω is an arbitrary flat form and both ψ and div ψ are integrable .

3) v ∧ w = (−1)kl w ∧ v , whenever v ∈ ∧k V and w ∈ ∧l V . ) In addition, we have that ∧0 V = R , ∧1 V = V . It follows that if {e1 , . . , en } is a basis of V , then {ei1 ∧ . . ∧ eik : 1 ≤ i1 < · · · < ik ≤ n} is a basis of ∧k V . In particular, n . k Elements in ∧k V are called k-vectors of V . If V ∗ is the dual space of V , we write dim ∧k V = ∧k V := ∧k V ∗ , ∧∗ V := ∧∗ V ∗ . Thus, if {e∗1 , . . , e∗n } is a basis of V ∗ , dual to {e1 , . . , en }, then {e∗i1 ∧ . . ∧ e∗ik : 1 ≤ i1 < · · · < ik ≤ n} is a basis of ∧k V .

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1973, Year of the Humanoids: An Analysis of the Fall UFO Humanoid Wave by David Webb

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