section 2.11 rewrite Oct 31

installed on Oct 31 at 5:30 PM 2.11 The dual spaces V* , V*V* and V*k. In our treatment of the dual space objects we shall attempt to use Greek or script letters for all dual space vectors and tensors encountered. Vectors in V will continue to be represented by Latin letters. Just as we used V as our prototype vector in space V, here we use V as our prototype vector in space V* . The dual space V* is by definition the space of linear functionals over V. If V ϵ V*, we can then write V : V → K V(v) = k ϵ K (2.11.1) where K is any field such as the reals. Since V is a linear functional, V(v) is a linear function. In normal calculus, if f: V → R, one refers to f as a function, and f(v) as that function evaluated at some point in V, though loosely speaking f(v) is also called a function. To emphasize the distinction, we shall refer to f as a "functional" and f(v) as a "function". The basis λi Given n vectors {ei} which are a basis for V, one can find another set of n basis vectors {ei} in V such that, ei ej = δij . (2.3.2) (2.11.2) These are the same ei discussed in Section 2.3. Section 2.7 gave an example of computing the ei from the ei. We can then define a basis {λi) for V* as a set of linear functionals λi such that λi(v) = ei v . // = Σa(ei)a va (2.11.3) Notice that i is a label, not a component. Functional λi is manifestly linear since λi(kv) = kλi(v) and λi(v + v') = λi(v) + λi(v') k ϵ K . (2.11.4) Here v and v' are vectors in V. As noted earlier, the definition of linearity is often combined into a single statement, λi(k1v + k2v') = k1λi(v) + k2λi(v') k1, k2 ϵ K . (2.11.5) The vectors ei are sometimes called "covectors" or "reciprocal vectors" or "dual vectors" which form a "dual basis". From (2.11.3) and (2.3.2) one then has λi(ej) = ei ej = δij . (2.11.6) Therefore, λi(v) = λi(Σj vj ej) = Σj vjλi(ej) = Σj vjδij = vi . (2.11.7) The linear functional λi applied to vector v thus gives the e-basis component vi of v. Comments: 1. The basis functionals λi can be defined directly from λi(ej) = δij without use of the dot product. Note that (2.11.7) λi(v) = vi does not depend on the existence of a dot product. The dot product implies that the vector space V is also a Hilbert Space and one need not assume this fact, but in our applications V will always be a Hilbert Space. 2. The covector ei is associated with the basis functional λi but one should not identify λi = ei. For one thing, λi is a scalar-valued functional while ei is a vector. One sometimes sees λi(v) written as ei(v) or as e*i(v), but we use λi to emphasize the distinction between λi and ei. 3. It is not hard to show that λi is a basis for V*. The proof relies on the fact that, since ei form a basis for V, the covectors ei also form a basis for V. 4. Some authors use notation v*i(v) in place of our λi(v) and vi in place of our ei. General vector in V* A general linear functional V in V* can be written as a linear combinations of the basis functionals λi, where the coefficients ai form a vector a in V, V = Σiaiλi = general vector in V* V(v) = Σiaiλi(v) V: V → K (2.11.8) On the right we show the corresponding function V(v). Evaluating at v = ej, V(ej) = Σiaiλi(ej) = Σiaiδij = aj // using (2.11.6) (2.11.9) so V(ei) picks off the coefficient ai appearing in the expansion (2.11.8). Consider now, V(v) = Σiaiλi(v) = Σiai vi = a v = Σiaivi . (2.11.10) Imagine a version of Picture A (2.1.1) where we transform from x-space with basis vectors ei to some x'-space with basis vectors e'i. Under such a transformation, a dot product like a v transforms as a scalar. So, whereas v and a transform as vectors, the function V(v) transforms as a scalar field, just as in (2.2.6). That is to say, V(v) = a v = a' v' = V'(v') (2.11.11) We see above that there is one-to-one mapping (an isomorphism) between the set of vectors a in V and the set of linear functionals V in V*. For this reason, V and V* have the same dimension n and the λi really do form a basis for V*. Although the function V(v) transforms as a scalar, the associated vector a (covector) transforms as a rank-1 tensor (vector), and it is through this isomorphic connection that we loosely refer to the function V(v) as being a rank-1 tensor. When functional V is expanded on the basis functionals λi, the coefficients ai are the rank-1 tensor, and these coefficients are projected out by (2.11.9). Rank-2 tensor in V*V* We want the object λiλj to be a linear functional over the space V*V* such that λiλj: V*xV* → K . (2.11.12) The natural way to accomplish this desire is to write (λiλj)(v1,v2) = λi(v1)λj(v2) = scalar * scalar = scalar ϵ K (2.11.13) The function (λiλj)(v1,v2) is bilinear in its arguments because λi(v1) is linear in its argument. The most general bilinear functional M in V*V* can be expanded in the following manner, analogous to (2.10.2) M = Σab [M(λ)]ab λaλb (2.11.14) where the coefficients [M(λ)]ab are assumed to transform as a covariant rank-2 tensor. Again, due to the isomorphism between the set of such rank-2 tensors and the set of bilinear functionals M, we loosely refer to the functional M as a rank-2 tensor, or just a 2-tensor. One may evaluate the bilinear functional M at (v1,v2) to get M(v1,v2) = Σab [M(λ)]ab (λaλb)(v1,v2) = Σab [M(λ)]ab λa(v1) λb(v2) . (2.11.15) Notice that the function M(v1,v2) is bilinear in its vector arguments because λa(v1) and λb(v2) are each linear. Evaluation at (e1,e2) yields M(ei,ej) = Σab [M(λ)]ab λa(ei) λb(ej) = Σab [M(λ)]ab δaiδbj = [M(λ)]ij . (2.11.16) This shows that the coefficients [M(λ)]ij of the bilinear functional M can be obtained from the function M(v1,v2) evaluated at the basis vectors (ei,ej), similar to what happened in (2.11.9). One can rewrite M(v1,v2) from (2.11.15) using (2.11.7) as M(v1,v2) = Σab [M(λ)]ab λa(v1) λb(v2) = Σab [M(λ)]ab (v1)a(v2)b . (2.11.17) Since this is a fully contracted outer product of three tensors (Section 3.1), one sees that M(v1,v2) transforms as a scalar field over V x V, M'(v'1,v'2) = M(v1,v2) . (2.11.18) Once again, although M transforms as a scalar, it is referred to as a 2-tensor because its expansion coefficients form a rank-2 tensor. As a special case, consider M = α β . We find that (α β)(v1,v2) = Σabαaβb (v1)a(v2)b = [Σaαa (v1)a][Σbαb (v1)b] = α(v1)β(v2) (2.11.19) Again, α β is called a 2-tensor even though (α β)(v1,v2) is a scalar. Notice from (2.11.19) and (2.11.9) that (α β)(ei,ej) = α(ei)β(ej) = αiβi = (α β)ij . (2.11.20) Warning: In the above equations, the symbols α and β are overloaded. For example, α represents a linear functional, while α is a vector (rank-1 tensor) whose components αi are the coefficients in the expansions of the linear functional α = Σiαiλi and of the linear function α(v) = Σiαivi . Rank-k tensor in V*V* ..... A rank-k tensor (k-multilinear functional) M in V*k ≡ V*V*.... can be expanded as in (2.10.14), M = Σabc... [M(λ)]abc... (λaλbλc ...) (2.11.21) with M(v1,v2, ....vk) = Σabc... [M(λ)]abc... λa(v1) λb(v2) λc(v3).... (2.11.22) The function M(v1,v2, ....vk) is thus seen to be k-multilinear in its k vector arguments. Using (2.11.7) one may write (2.11.22) as, M(v1,v2, ....vk) = Σabc... [M(λ)]abc... (v1)a(v2)b(v3)c... (2.11.23) which shows by full contraction that M(v1,v2, ....vk) is a scalar field over Vk , M'(v'1,v'2, ....v'k) = M(v1,v2, ....vk) . (2.11.24) The coefficients in (2.11.21) can be obtained from M(ei,ej,ek ....) = Σabc... [M(λ)]abc... λa(ei) λb(ej) λc(ek)... = Σabc... [M(λ)]abc... δai δbj δck .... = [M(λ)]ijk... For the special case where M = α β δ ..... , (α β δ .....)(v1,v2, ...vk) = α(v1)β(v2)δ(v3)...... . (2.11.25) Once again, due to the isomorphism between functionals M and functions M(v1,v2, ....vk) , one loosely refers to the scalar field M(v1,v2, ....vk) as a rank-k tensor, although M is the actual rank-k tensor object whose components [M(λ)]abc... transform by the usual rule. One might say that the rank-k tensor M is "represented by" the k-multilinear function M(v1,v2, ....vk) , just as it is represented by [M(λ)]abc... in the sense of tensor components.