Overview

This is the Title PhL 1.27.16 This has been installed, do not edit. Overview This document is meant as a sort of User Guide for both tensor products and wedge products. These objects are often glossed over in literature that makes heavy use of them, the assumption being that everything is obvious and not worth describing too much. Our monograph is very focused on these two kinds of products and has little to say about applications such as quantum mechanics, differential forms, the generalized Stokes's Theorem or supersymmetry. We attempt to include both the mathematical view and the engineering/physics view of things, but the emphasis is on the practical. The notation is entirely covariant, which one can say includes the practical for Cartesian spaces where indices can be all down. The study of wedge products is known as the exterior algebra and is credited to Grassmann. Summary Chapter 1 surveys the mathematician's description of the tensor product as a quotient space, and then places the tensor product in the framework of category theory. This approach is resumed much later in Chapter 9 for the wedge product, after the reader is more familiar with that object. Chapter 2 reviews tensor analysis and then introduces a meaning for the tensor product symbol in terms of outer products of tensors. After a quick review of tensor expansions and projections, the last section introduces the notion of a dual space and includes the use of the Dirac bra-ket notation. The notion of a tensor function is introduced. Chapter 3 discusses the theory of Chapter 1 versus the practicality of Chapter 2 in terms of outer products. It then derives the Kronecker product of two matrices in covariant notation. This topic is somewhat tangential to the main development, but is included since it is often not explained very well in the literature. Maple is used to compute a few such Kronecker products. Chapter 4 has four parts involving products of two vectors and their vector spaces: tensor product, dual tensor product, wedge product, and then dual wedge product. Chapters 5,6,7,8 continue this order of presentation but for products of k vectors and then of general tensors. The order is : tensor product (Ch 5), dual tensor product (Ch 6), wedge product (Ch 7) and then dual wedge product (Ch 8). The chapters intentionally have a high degree of parallelism, though details are omitted from the later chapters to reduce repetition. It is the dual tensor chapters which involve tensor functions as the closure of tensor functionals onto a general set of vectors. The tensor-product tensor functions are multilinear, whereas the wedge-product ones are multilinear and totally antisymmetric. Alternate wedge product normalizations are discussed. Chapter 9 returns to the mathematician's world giving a description of the wedge product in terms of quotient spaces. Appendix A explains our permutation notation and the powerful rearrangement theorem used in various proofs throughout the document. The Alt and Sym operator properties are presented in a generic permutation space, and then those generic results are applied to tensors and tensor functions. The permutation tensor ε is given honorable mention, and a few obscure theorems are proved. Appendix B discusses the direct sums of vectors, vector spaces and operators in those spaces. Appendix C shows that when one antisymmetrizes a product of tensors, pre-antisymmetrizing one or more of those tensors makes no difference in the result. Appendix D shows how tensors and tensor functions are the same objects expressed in different bases.