retired section 4

This is the Title PhL 1.11.15 Note that page numbering is turned on in this template and view is 125%, located in phil/roaming/microsoft/templates size about 219K. retired section 4, do not edit 4. The wedge product in the direct product space V* x V* We follow the approach of the Section 3 for the direct product space V x V, but now we have V* x V*. As found earlier, the wedge product only makes sense if W = V, and our dual-space wedge product only makes sense if W* = V*. So for the basis functionals {λi} of V* we define the following wedge product which exists in the direct product space V* x V* = V*2, λi^ λj ≡ λiλj - λjλi where we have antisymmetrized on the two basis functionals. The definition implies that λi ^ λj = - λj ^ λi and λi ^ λi = 0 . Like λiλj, λi^ λj is a bilinear functional in the space V* x V* = V*2, and we can evaluate the functional at a point (v1,v2) = v1v2 in the V * V = V2 direct product space to obtain (λi^ λj)(v1,v2) = (λiλj)(v1,v2) – (λjλi)(v1,v2) = λi(v1) λj(v2) – λj(v1) λi(v2) = det . In the above, (λi^ λj)(v1,v2) indicates a bilinear function whose name is (λi^ λj) which is evaluated at the point (v1,v2) in V2. Notice that (λi^ λj)(v1,v2) is antisymmetric under i ↔ j, and is also antisymmetric under v1 ↔ v2. Either of these interchanges swaps the columns of the determinant's matrix. As before, the scalar and distributive rules of are transferred to ^ . Thus, if α and β are two general linear functionals in V*, we have α = Σiaiλi β = Σjbjλj α β = ( Σiaiλi) ( Σjbjλj) = Σijaibj(λ i λj) . Tracing the same logic path as used for a ^ b above, one sees that that α ^ β = (Σiaiλi) ^ ( Σjbjλj) = Σijaibj (λi^λj) // rules for ^ above = Σi<j [ aibj - ajbi] (λi^λj) = Σi<j det (λi^λj). From this result, we obtain these general rules for the wedge product of two vectors in V* (α ^ β) = - (β ^ α) (α ^ α) = 0 which are just the rules for the basis vectors λi translated to general vectors α and β, Recall that here α ^ β is a bilinear functional (in the space V*2) which acts on elements of V2. Thus we can evaluate the above functionals at a point (v1,v2) = v1v2 in V2 to obtain (α ^ β)(v1,v2) = Σi<j [ aibj - ajbi] (λi^λj)(v1,v2) = Σi<j [ aibj - ajbi] [λi(v1) λj(v2) – λj(v1) λi(v2)] = Σi<j [ aibj - ajbi] λi(v1) λj(v2) – Σi<j [ aibj - ajbi] λj(v1) λi(v2)] = Σi<j [ aibj - ajbi] λi(v1) λj(v2) – Σi>j [ ajbi - aibj] λi(v1) λj(v2)] = Σi<j [ aibj - ajbi] λi(v1) λj(v2) + Σi>j [ aibj - ajbi] λi(v1) λj(v2)] = Σi≠j [ aibj - ajbi] λi(v1) λj(v2) = Σi,j[ aibj - ajbi] λi(v1) λj(v2) One can tell by inspection of this last expression that (β ^ α)(v1,v2) = – (α ^ β)(v1,v2) (α ^ β)(v2,v1) = – (α ^ β)(v1,v2) (α ^ β)(v1,v2) is a bilinear function of v1 and v2 (separately linear in each variable). The first result is obvious by taking ak ↔ bk . The second follows from the simple fact that rename i↔j Aji = - Aij reorder factors Σij Aijfi(x)fj(y) = Σij Ajifj(x)fi(y) = - Σij Aijfj(x)fi(y) = - Σij Aijfi(y)fj(x) where Aij is any antisymmetric matrix. The last fact follows since λi(v1) is a linear functional in v1 and λj(v2) in v2. One can say that the wedge product of any two linear functionals of V* ( like α and β) when evaluated at the point (v1,v2) in V2 forms an antisymmetric bilinear function of (v1,v2). The space of such functions is often called Λ2(V), but we can also identify Λ2(V) with the space of all antisymmetric bilinear functionals acting on V2. In other words, if α,β ϵ V then α β ϵ V*2 but α ^ β ϵ Λ2(V) and Λ2(V) V*2. In the notation Λ2(V), the Λ stands in effect for Antisymmetric, the superscript 2 is the number of vectors being wedged together, and V is the space on which the linear functionals α and β act.