scraps4

Since ek is itself a vector in V, one can write ek = Σi (ek)iei . Notice that k is a label on ek, whereas the i on (ek)i is a component index. So the components of the vector ek in the ei basis are (ek)i. In Section 2.6 we expended some effort clarifying this little expansion, see (2.6.5) and (2.6.6). Given any two vectors v1 and v2 in V, consider the dot product (2.2.5) (no longer bolding vectors), f(v1,v2) ≡ v1 v2 = Σk=1n(v1)k(v2)k (4.1.2) which clearly is an element of the field K. This function f: VxV → K is manifestly bilinear in its two vector arguments, see (1.1.6). This sum can of course be applied to any two basis vectors, so that ei ej = Σk=1n(ei)k(ej)k (4.1.3) We now make two definitions: The {ei} are orthogonal if ei ej = fiδi,j (4.1.4) The {ei} are orthonormal if ei ej = δi,j (4.1.5) **************************************** {λi} = basis of V* {λi'} = basis of W* . (4.2.1) Like all elements of V*, the object λi ϵ V* is also a linear functional over V. This is a linear functional which, when evaluated at a point in V, produces a scalar: λi : V → K. As usual, we think of K as the reals R, but we try to stay more general by having K be an arbitrary field. 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(x) as a "function". Given the 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) (4.2.2)  The general method of finding the ei from the ei was given in Section 2.7 with an example. The action of the linear functional λi can then be taken as λi(v) = ei v (4.2.3) which is manifestly linear since λi(kv) = kλi(v) and λi(v + v') = λi(v) + λi(v') k ϵ K . (4.2.4) 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 (4.2.5) The vectors ei are sometimes called "covectors" or "reciprocal vectors" or "dual vectors" which form a "dual basis". From (4.2.3) one then has λi(ej) = ei ej = δij . (4.2.6) The motivation for selecting ei as the covector for λi in (4.2.3) was exactly to obtain result (4.2.6). General linear functionals in V* and W* can be written as linear combinations of the basis functionals, α = Σiαiλi = general vector in V* α(v) = Σiαiλi(v) α: V → K β = Σjβjλj' = general vector in W* β(v) = Σjβjλj'(v) β: W → K (4.2.7) where on the right we show the corresponding functions α(v) and β(v). Using (4.2.6) one sees that evaluation at v = ej gives α(ej) = Σiαiλi(ej) = Σiαiδij = αj and similarly for β, so α(ei) = αi β(e'j) = βj . (4.2.8) There is a dual tensor product space V*W* which has the basis λiλ'j . (4.2.9) We want this object λiλ'j to be a functional over the space V*W* such that λiλ'j: V*xW* → K . (4.2.10) The natural way to accomplish this desire is to write (λiλ'j)(v,w) = λi(v)λj'(w) = scalar * scalar = scalar ϵ K (4.2.11) This function is manifestly "bilinear" in that it is separately linear in each of the arguments v and w. For example, (λiλ'j)(v1+v2,w) = λi(v1+v2)λj'(w) = [λi(v1) + λi(v2)]λ'j(w) = λi(v1)λj'(w) + λi(v2)λj'(w) = (λiλ'j)(v1,w) + (λiλ'j)(v2,w) . (4.2.12) The "rules" 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, (kα) β = α (kβ) = k (α β) k 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 (4.2.13) In analogy with (4.1.9), a general element of the dual tensor product space V*W* can be written T ≡ Σij Tij λiλ'j T ϵ V*W* . (4.2.14) where Tij ≡ [T(λ,λ')]ij are coefficients in the field K. 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) ϵ K (4.2.15) so one may regard T : VxW → K. Setting v = ei and w = e'j and using ***, one finds that T(ei,e'j) = Tij ϵ K (4.2.16) which provides an interpretation of Tij as the function T(v,w) evaluated at two basis vectors ei,e'j. This may be compared with α(ei) = αi. More specifically, consider α ϵ V*and β ϵ W* as shown above. Then α β = (Σiαiλi)(Σjβjλ'j) = Σij αiβj λiλ'j ϵ V*W* (4.2.17) (α β)(v,w) = Σij αiβj (λiλ'j)(v,w) = Σij αiβj λi(v)λj'(w) ϵ K (4.2.18) which then is just a particular example of (4.2.15). Another way to write the above line is (α β)(v,w) = Σij αiβj λi(v)λj'(w) = (Σiαi λi(v))(Σjβj λ'j(v)) = α(v) β(v) (4.2.19) In particular (α β)(ei,e'j) = α(ei) β(e'j) = αiβj . (4.2.20) If it happens that W = V, then W* = V* and we write V*W* = V*V* = V*2. Then the most general element T of V*2 can be expressed in analogy with (4.1.10) using Tij ≡ [T(λ)]ij, T = ΣijTij (λiλj) (4.2.21) T(v1,v2) = Σij Tij (λiλj)(v1,v2) = Σij Tij λi(v1)λj(v2) (4.2.22) T(ei,ej) = Tij . (4.2.23) One says that the linear functionals (α β) and T in the dual space V*2 are "tensors" of rank 2. Whereas T earlier was a rank-2 tensor in V2, this object T is a rank-2 tensor in the dual space V*2. In the abstract, it is T which is the real tensor and it is shown expanded on a particular basis {λi} which in turn is associated with a particular basis {ei} of V as discussed above. What are the "components" of tensor T ? In one sense they are the coefficients [T(λ)]ij of the expansion shown above onto the basis (λiλj), in analogy with the non-dual space. But in the dual space this is not the sense of "component" we are interested in. In the dual world, the function vector arguments play the role that vector component indices play in the outer products of the non-dual world. For example, compare these equations: (a b)ij = aibj (2.8.10) or (3.1.8) // VW (α β)(v,w) = α(v)β(w) (4.2.19) // V*W* (4.2.24) T = Σij Tij eie'j (4.1.9) // VW T = Σij Tij λiλ'j (4.2.14) // V*W* (4.2.25) Tab = ΣijTij (eie'j)ab = Σij Tij (ei)a(e'j)b // VW T(v,w) = Σij Tij (λiλ'j)(v,w) = ΣijTij λi(v)λ'j(w) . // V*W* (4.2.26) So it is really the T(v,w) which are the "tensor components" of the tensor T, whereas the Tij are the coefficients in the above expansion, In VW the "tensor components" are Tij . One can then regard T(v,w) as the tensor of interest in terms of its components. For example, Spivak on page 75 refers (in effect) to our function T(v1,v2) as a being a 2-tensor. *********************** What are the "components" of tensor T ? In one sense they are the coefficients [T(λ)]ij of the expansion shown above onto the basis (λiλj), in analogy with the non-dual space. But in the dual space this is not the sense of "component" we are interested in. In the dual world, the function vector arguments play the role that vector component indices play in the outer products of the non-dual world. For example, compare these equations: (a b)ij = aibj (2.8.10) or (3.1.8) // VW (α β)(v,w) = α(v)β(w) (4.2.19) // V*W* (4.2.24) T = Σij Tij eie'j (4.1.9) // VW T = Σij Tij λiλ'j (4.2.14) // V*W* (4.2.25) Tab = ΣijTij (eie'j)ab = Σij Tij (ei)a(e'j)b // VW T(v,w) = Σij Tij (λiλ'j)(v,w) = ΣijTij λi(v)λ'j(w) . // V*W* (4.2.26) So it is really the T(v,w) which are the "tensor components" of the tensor T, whereas the Tij are the coefficients in the above expansion, In VW the "tensor components" are Tij . One can then regard T(v,w) as the tensor of interest in terms of its components. For example, Spivak on page 75 refers (in effect) to our function T(v1,v2) as a being a 2-tensor.