Sec 4_1 and 4_2 scratch

Scratch doc for Sections 4.1 and 4.2 Have to fix things up here. 4. The Wedge Product of 2 vectors built on the Tensor Product We now back up and reconsider the space VW and its elements vw. The goal of the next two sections is to establish the parallelism between the vector space VW and the "dual" vector space V*W*. Some repetition is used to review and reinforce earlier stated facts. Then Sections 4.3 and 4.4 introduce the wedge product developed in a similar parallel fashion. At the end of each of the four sections below a selection of equations is reexpressed in Dirac notation. 4.1 The tensor product of 2 vectors in V2 We now drop the bolding of vectors to simplify notation and to be compatible with Chapter 1. Let en be an arbitrary basis for vector space V (x-space, dim n) where, as in (2.5.9), we interpret these en as the tangent base vectors of some transformation x' = F(x) with differential dx' = Rdx. The expansion of vector v is given by v = Σi=1n v'iei as in (2.7.1) line 3, and the vector transformation rule is v'i = Rijvj (v' = Rv) as in (2.1.2). Let en be an arbitrary basis for vector space W (x-space, dim n) where, as in (2.5.9), we interpret these en as the tangent base vectors of some transformation x' = F(x) with differential dx' = Rdx. The expansion of vector w is given by w = Σj=1n w'jej as in (2.7.1) line 3, and the vector transformation rule is v'i = Rijvj (w' = Rw) as in (2.1.2). We then have: {ei} = basis of V dim(V) = n v = Σi=1n v'i ei = general vector in V v'i ϵ K {ei} = basis of W dim(W) = n w = Σj=1n w'j ej = general vector in W w'j ϵ K v' = Rv w' = Rw differentials for different transformations x' = F(x) and x' = F(x) ei = |ei> ei = |ei> // Dirac notation {eiej} = basis for the tensor product space VW dim(VW) = n*n vw = a pure "vector" in the tensor product space VW vw ≠ wv if v ≠ w : VxW → VW : (v,w) ↦ vw (4.1.1) The last line shows as a mapping → between two sets, while ↦ shows how set elements map. Note that vw ≠ wv. For V≠W, wv does not even make sense since that requires w ϵ V and v ϵ W. For V = W the objects vw and wv are still different unless v = w. Outer Product Revisited The notion of an outer product was discussed in Sections 2.8 and 3.1. We had for example (where ai and bj are the covariant components of vectors a and b both ϵ V), (a b)ij = aibj // outer product of two vectors (3.1.8) (A B)abcd = AabBcd . // outer product of two rank-2 tensors (3.1.10) The "outer product" of two vectors a and b may be written in vector/matrix notation as follows, (a b)** = abT = ( b1. b2....bn) = (a b)ij = (abT)ij (4.1.2) The same vector/matrix notation used above can also be used to express the "inner product" (dot product) appearing in (2.2.5), with the caveat noted below, a b = aTb = ( a1. a2....an) = Σk=1n akbk = <a | b> (4.1.3) If the a components are contravariant, the b components must be covariant, and vice versa. Chapter 1 Tensor Product Revisited By convention one represents an element of a tensor product space using the symbol. It is a certain kind of "product" between a vector in one vector space and a vector in another vector space. On can treat as an operator : VxW → (VW) in the sense that (v,w) = (v) (w) = (vw) = element of tensor product space (VW). Certain rules were declared in (1.1.5) which make the tensor product space be a vector space, and which in an intuitive sense just seem "reasonable", (sv) w = v (sw) = s (vw) // s = scalar (ϵ K) v (w1+ w2) = vw1 + vw2 // left distributive property (v1 + v2) w = v1w + v2w . // right distributive property (1.1.5) (4.1.4) In the last two equations, the + on the left represents addition in either W or V, whereas the + on the right side represents addition in VW. These lines say that multiplication "distributes" over addition +. The scalar rule can be combined with the distributive rules to obtain this equivalent rules restatement: v (s1w1+ s2w2) = s1(vw1) + s2(vw2) // s1,s2 = scalar (ϵ K) (s1v1 + s2v2) w = s1(v1w)+ s2(v2w) .// s1,s2 = scalar (ϵ K) (1.1.7) (4.1.5) The above rules in effect say that defines a "bilinear" operation -- it is linear separately in each of its operands. Notice that the following two rules are incorrect: v w = w v // wrong! (unless V = W and v = w) (sv) (sw) = (vw) // wrong! (unless s = 1) As noted in Appendix B the second rule applies to a direct sum . Using the correct "rules" above, one may write v w = ( Σiv'iei)( Σjw'jej) = Σijv'iw'j (eiej) (4.1.6) showing how this pure tensor product vector can be expressed in terms of the basis functions. In Dirac notation |v> |w> = Σijv'iw'j |ei> |ej> . (4.1.6a) General tensors in VW and V2. A general "vector" (rank-2 cross tensor) in WV can be written as a linear combination of the basis vectors, using the notation of (2.7.10), T ≡ Σij [T(e,e)]ij eiej T ϵ VW . Σij ≡ Σi=1nΣj=1n (4.1.7) If W = V, we refer to the space VW = VV as V2, and then T ≡ Σij T'ij eiej T ϵ VV = V2 . Σij ≡ Σi=1nΣj=1n (4.1.8) Although we have said T is a "vector" in the abstract sense that a vector space (even a tensor product vector space) has "vectors" as elements, the usual terminology is to say that T is a "rank-2 tensor" in the space V2. Meanings of tensor. The word "tensor" has a weak and a strong meaning. In the weak meaning, a rank-2 tensor is something that has components with two indices like Tij. In the strong meaning, a rank-2 tensor is a set of components Tij which transform in a certain manner with respect to some underlying transformation, T'ab = Raa' Rbb' Ta'b' Picture A T ϵ VV (2.1.6) Covariant expansion forms. The rank-2 tensor T can be expanded in any of these equivalent ways T ≡ Σij [T(e,e)]ij eiej T ϵ VW Σij ≡ Σi=1nΣj=1n T ≡ Σij [T(e,e)]ij eiej T ≡ Σij [T(e,e)]ij eiej T ≡ Σij [T(e,e)]ij eiej . (4.1.9) The forms differ only by reversing tilts as was described below (2.9.2). If W = V, then T ≡ Σij T'ij eiej T ϵ VV = V2 Σij ≡ Σi=1nΣj=1n T ≡ Σij T'ij eiej T ≡ Σij T'ij eiej T ≡ Σij T'ij eiej . (4.1.9) Danger: Now the prime has two separate meanings!!! Default Notation. In the rest of this document, unless otherwise specified, expansions in Vk shall always be on the ei basis, and the label (e) appearing for example on [T(e)]ab in (2.6.8) or [ei(e)]b in (2.6.4) will be omitted. For example, Tab = [T(e)]ab for T ϵ VV // Tab = [T(e,e')]ab for T ϵ VW (ei)b = [ei(e)]b = δib for V // (e'i)b = [e'i(e')]b = δib for W (eiej)ab = [(eiej)(e)]ab = [ei(e)]a [eb(e)]b = (ei)a (ej)b = δia δjb . (4.1.10) Dot Products. One can define a covariant dot product between two elements of V2 in this manner A B ≡ ΣijAijBij = ΣijAijBij = ΣijBijAij = B A . (4.1.11) If A or B is a pure rank-2 tensor, one can write as well (ab)B = ΣijaibjBij A(cd) = ΣijAijcidj (ab)(cd) = Σijaibjcidj = (ac)(bd) . (4.1.12) The last line appears as (2.9.13). Dirac Notation for Section 4.1 . It seemed best not to clutter that above text with these alternate forms. The Dirac version of an equation gets a D subscript on its equation number. ei = |ei> ei = |ei> bases for V and W eiej = |ei> |e > basis for the tensor product space VW vw = |v> |w> a pure "vector" in the tensor product space VW (4.1.1)D T = |T> = Σij [T(e,e)]ij |ei> |ej> rank-2 tensor in VW T = |T> = Σij T'ij |ei> |ej> rank-2 tensor in V2 (a b)ij = <ui| <uj| |a> |b> = <ui|a> <uj|b> = aibj outer product a b = <a | b> dot product |v> |w> = Σijviwj |ei> |ej> expansion of pure vector on basis vectors (4.1.6)D <ea| <eb| |T> = <ea| <eb| Σij [T(e,e)]ij |ei> |ej> = Σij [T(e,e)]ij <ea|ei><eb|ej> = Σij Tij δaiδbj = [T(e,e)]ab (2.10.5)D <a| <b| |c> |d> = <a|c><b|d> = Σiaici Σjbjdj = Σijaibjcidj dot product in V2 (4.1.12)D 4.2 The tensor product of 2 dual vectors in V*2 The dual space V* of V was discussed in Section 2.11. Space W* is dual to W. We continue our convention of using Greek or script letters for dual space objects. The current section is basically a generalization of Section 2.11 to the case where V and W are different vector spaces. We show corresponding Section 2.11 equations in italics. Basics. Consider the two dual vector spaces V* and W* (defined over field K) of dimension n and n'. Let {λi} = basis of V* dim(V*) = n α = Σi=1n α'i λi = general linear functional in V* {λi} = basis of W* dim(W*) = n β = Σj=1n β'j λj = general linear functional in W* λi = (ei)T = <ei| λi = (ei)T = <ei| // matrix and Dirac notation {λiλj} = basis for the tensor product space V*W* dim(V*W*) = n*n αβ = a pure "vector" in the tensor product space V*W* αβ ≠ βα if α ≠ β : V*xW* → V*W* : (α,β) ↦ αβ (4.2.1) Note that αβ ≠ βα. For V*≠W*, βα does not even make sense since that requires β ϵ V* and α ϵ W*. For V* = W* the objects αβ and βα are still different unless α = β. Vector expansions in V* and W*. Linear functionals in V* and W* can be written as linear combinations of the basis functionals, α = Σiαiλi α(v) = Σiαiλi(v) α: V → K (2.11.8) β = Σjβjλ'j β(v) = Σjβjλj'(v) β: W → K . (4.2.2) The middle column shows the corresponding functions α(v) and β(v), and we now have α(ei) = αi (2.11.9) β(e'j) = βj . (4.2.3) Basis tensors in V*W*. The basis functionals for V*W* are the λiλ'j where, (λiλ'j)(v,w) = λi(v)λ'j(w) = scalar * scalar = scalar ϵ K (2.11.15) (4.2.4) where (recall these are called the "ith coordinate functions") λi(v) = vi λ'i(w) = wi (2.11.7) (4.2.5) so that (λiλ'j)(v,w) = viwj . (2.11.16) (4.2.6) This function is manifestly bilinear in its two vector arguments. The Rules for V*W* . The "rules" (1.1.5) for the operator in the space V*W* are the same as those for in the space VW, since V*W* is, after all, a tensor product of two spaces, (sα) β = α (sβ) = s (α β) s ϵ K, α ϵ V* β ϵ W* α (β1+ β2) = α β1 + α β2 // distributive property (α1 + α2) β = α1β + α2β . // same idea as above (4.1.4) (4.2.7) Rank-2 cross-tensor expansion in V*W*. A general functional of the dual tensor product space V*W* can be written T ≡ Σij Tij λiλ'j T ϵ V*W* Σij ≡ Σi=1nΣj=1n' (2.11.17) (4.2.8) where Tij ≡ [T(e,e')]ij are coefficients in the field K. The Tij here are exactly the same Tij which appear in the VW expansion (4.1.7), T = ΣijTij eie'j . Evaluating at a point (v,w) in VxW one gets, T(v,w) = Σij Tij (λiλ'j)(v,w) = Σij Tij λi(v) λj'(w) = Σij Tij viwj (2.11.18) (4.2.9) so one may regard T : VxW → K, and T(v,w) as manifestly bilinear in its arguments. Setting v = ei and w = e'j , one finds that T(ei,e'j) = Tij ϵ K (2.11.19) (4.2.10) This may be compared with α(ei) = αi in (4.2.3). An arbitrary rank-2 tensor functional T can be represented either by its expansion T = ΣijTijλiλ'j or by the corresponding tensor function T(v,w). As a special case, consider α ϵ V*and β ϵ W* as shown above. Then, α β = (Σaαaλa)(Σbβbλ'b) = Σab αaβb λaλ'b ϵ V*W* (2.11.21) (4.2.11) (α β)(v,w) = Σabαaβb(λaλ'b)(v,w) = Σabαaβbλa(v)λb'(w) = Σabαaβbvawb (2.11.22) (4.2.12) which then is just a particular example of (4.2.9). Continuing the above, (α β)(v,w) = Σabαaβbvawb = [Σaαava][Σbαbwb] = α(v) β(w) . (2.11.22) (4.2.13) In particular, (α β)(ei,e'j) = α(ei) β(e'j) = αiβj = (α β)ij . (2.11.23) (4.2.14) If it happens that W = V, then W* = V* and we write V*W* = V*V* = V*2. The equations above then revert to those given in Section 2.11 (referenced in italics above). Dirac Notation for Section 4.2 λi = (ei)T = <ei| λ'i = (e'i)T = <e'i| bases for V* and W* λiλ'j = <ei| <e'j| basis for the dual tensor product space V*W* αβ = <α| <β| pure element of V* W* (4.2.1)D α = <α| = Σiαi<ei| α(v) = <α|v> = Σiαi<ei|v> vector functional expansions β = <β| = Σiβi<e'i| β(v) = <β|v> = Σiβi<e'i|v> (4.2.2)D α(ei) = <α|ei> = αi vector function α(v) evaluated at v = ei β(ei) = <α|ei> = βi vector function β(v) evaluated at v = ei (4.2.3)D (λiλ'j)(v,w) = <ei| <e'j| |v> |w> = <ei| v><e'j|w> = viwj (4.2.4)D (4.2.5)D T = <T| = ΣijTij<ei | <e'j| = Σij Tij λi λ'i rank-2 tensor functional in V*W* (4.2.8)D T(v,w) = <T | |v> |w> = <T |v,w> rank-2 tensor function for V*W* = ΣijTij<ei | <e'j| |v> |w> = ΣijTij<ei |v> <e'j|w> = ΣijTijviwj (4.2.9)D (α β)(v,w) = <α | <β| |v> |w> = <α | v> <β | w> (4.2.13)D The vector spaces V2* and V2*f We can regard both T = <T| and T(v,w) = <T|v,w> as representations of the same rank-2 tensor functional <T| in V*W*. The object T is a bilinear rank-2 tensor functional, whereas T(v,w) is a bilinear rank-2 tensor function (a Spivak 2-tensor). There is a 1-to-1 correspondence between T and T(v,w) . We shall say T ϵ V*W* while T(v,w) ϵ (V*W*)f (f = function), and the two spaces are isomorphic. If W = V, then T ϵ V2* and T(v,w) ϵ V2*f and V2* and V2*f are isomorphic. Fact: The vector space V*2 is equivalent to the vector space V*2f of bilinear functions on V2. (4.2.15)