outline of opening section REVIEWED

Outline of Opening Section PhL 9.23.15 5.16.16 an early outline of early sections. 1. The Tensor Product 1 1.1 The Tensor Product as a Quotient Space 1 1.2 The Tensor Product in Category Theory 5 1.3 Outer Products 7 1.4 Tensor products of the form VV...V : The Tensor Algebra 9 1.5 Kronecker Products 12 Opening text nomenclature tensor product versus direct product 1.1 The Tensor Product as a Quotient Space Cartesian product VxW F(VxW) space (not called free vector space), pure and mixed elements equivalence relations / classes in space N VW ≡ F(VxW)/N = the quotient idea *** the end result (practical meaning of VW) *** the rules are bilinear, similar to bilinear functions *** basis and general linear combinations versus pure elements in VW *** vector versus tensor. *** product of dimensions rule F(VxWx...xZ)/N = VW...Z and k-multilinear *** 1.2 The Tensor Product in Category Theory category diagrams and history of category theory arrows, morphisms, functors. function composition. Roman reference, and the triangle diagram must commute g factored through f ; homF(V,W; X) and homomorphism construct τ as a linear function f and g are then a universal pair, τ a mediating morphism how the rules arise in this approach, again we have k-multilinear generalization to VW...Z and Lang reference nada 1.3 Outer Products summary of the tensor product so far components as a "extension" of the tensor product, why it works expansion of a rank-2 tensor and its components in simple and general basis *** how state things in covariant notation such as en and en outer product of 3 vectors, then of k vectors 1.4 Tensor products of the form VV...V : The Tensor Algebra interest now is in Vk meanings of the word "tensor" not all tensors can be written as outer products tensor product of two rank-2 tensors makes a rank-4 tensor special comma notation mentioned in passing other examples of outer products the general tensor with multiindex notation components direct sum concept the tensor algebra with its "mixed" elements of mixed grades graded algebra explained rank rule for of two mixed tensors in T 1.5 Kronecker Products two operators S and T and their spaces and non-square matrices rule for (ST)(vw) makes this be a bilinear thing exercise computing (ST) acting on a general rank-2 tensor the comma notation object (ST)ii',jj' really just rank-4 tensor arranging as (xy)ii' = Σjj' (ST)ii',jj' (vw)jj' then arranging as q'r = Σs Mrs qs M is the Kronecker product written symbolically as M = ST. two examples given, using simple Maple code 2.1 The tensor product of 2 vectors. statement of basis functions discussion of basis functions: orthogonal and orthonormal without using dot product outer product revisited, nmatrix notation inner product stated but not called inner product tensor product revisited the rules stated both ways the wrong rules! General element T with Fijcoefficients word tensor, and examples with rotation matrices contravariant and covariant forms with special ei 2.2 The tensor product of 2 dual vectors. statement of basis for V* and W* introduction of the qi vectors, and derivation that λi(ej) = qiTej = δi,j . statement of general linear functionals in V* and W* evaluation of α(ei) as αi first mention of the λiλ'j basis, and its mapping the function (λiλ'j)(v,w) introduced, show it is bilinear the rules for V*W* general element T with Fij coefficients function form of same expansion fact that T(ei,e'j) = Fij α β and (α β)(v,w) and (α β)(ei,e'j) = α(ei) β(e'j) = αiβj restate several results for the case W = V. comment on the "components" of T a final Fact stated 2.3 The wedge product of 2 vectors (a) definition of wedge and about L2 definition a ^ b ≡ ab - ba and simple derived rules statement of the ^ rules with some derivation general element T with Fij coefficients example with Fij = aibj partition of the basis functions into two sets (b) how big is L2 compared to V2? show T = Σij Fij (ei ^ ej) = Σi<j Aij (ei ^ ej) counting the elements of L2 and V2 statement of ratio of element counts, and alternate derivation (c) Wedge products and determinants: the geometry connection. write a^b as a determinant with i<j summation graphic showing what the determinants are two geometric connections the Hodge duality and the cross product connection 2.4 The wedge product of 2 dual vectors same definition, but uses α and β as vectors the ^ rules restated general element T with Fij coefficients example with Fij = αiβj show T = Σij Fij (λi ^ λj) = Σi<j Aij (λi ^ λj) counting the elements of Λ2 and V*2