In writing this book the intention is to assist the student in the learning process for structural and finite element analysis, in which he must master, not only the numerical techniques available, but also the mathematical skills necessary for the efficient description of the physics of the problem at hand. As such it is a teaching book. The format, then, is first to present a discussion on mathematical preliminaries in which the concepts of vectors, tensors and matrices are introduced. Which notation should be used? Vector, tensor or matrix? The answer is not simple, and the three notations are certainly not mutually exclusive. Following a discussion of the merits of the various notations, the reader is introduced to the ubiquitous Gauss’ divergence theorem. This theorem forms the basis of most descriptions of the integral of the rate of change of a variable over a region to its values on the surface of the region. In structural mechanics the relationship has been discovered independently and given such names as principle, of virtual displacements, principle of virtual forces, or simply virtual work. The advantage of the more general formulation by Gauss is in its adaptability to a variety of other physical situations where the concepts of static equilibrium are not available. In structural mechanics it is possible to extend the concept of virtual displacements (forces) to the contragredient principle which succinctly describes the relationship between statically equivalent force systems and their compatible displacements. The contragredient principle can be loosely described as a reflective principle (for the purposes of memory only). That is, if the force systems (P, Q) are connected through the relationship, https://www.w3.org/1998/Math/MathML"> { P } = [ B ] { Q } https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203867075/fe06687f-41e9-4cc2-adf0-8585ee49d327/content/eq1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>