In order to make this book sufficiently self-contained, it is necessary to include this chapter dealing with the mathematical tools that are needed to achieve a complete understanding of the topics discussed in the remaining chapters. Vector and tensor algebra is discussed, as is the important concept of the general directional derivative associated with the linearization of various nonlinear quantities that will appear throughout the book.
Readers, especially with engineering backgrounds, are often tempted to skip these mathematical preliminaries and move on directly to the main text. This temptation need not be resisted, as most readers will be able to follow most of the concepts presented even when they are unable to understand the details of the accompanying mathematical derivations. It is only when one needs to understand such derivations that this chapter may need to be consulted in detail. In this way, this chapter should, perhaps, be approached like an instruction manual, only to be referred to when absolutely necessary. The subjects have been presented without the excessive rigors of mathematical language and with a number of examples that should make the text more bearable.
VECTOR AND TENSOR ALGEBRA
Most quantities used in nonlinear continuum mechanics can only be described in terms of vectors or tensors. The purpose of this section, however, is not so much to give a rigorous mathematical description of tensor algebra, which can be found elsewhere, but to introduce some basic concepts and notation that will be used throughout the book. Most readers will have a degree of familiarity with the concepts described herein and may need to come back to this section only ifclarification on the notation used further on is required. The topics will be presented in terms of Cartesian coordinate systems.
Boldface italic characters are used to express points and vectors in a three-dimensional Cartesian space. For instance, e1, e2, and e3 denote the three unit base vectors shown in Figure 2.1, and any given vector v can be expressed as a linear combination of these vectors as
In more mathematical texts, expressions of this kind are often written without the summation sign, Σ, as
Such expressions make use of the Einstein or summation convention, whereby the repetition of an index (such as the i in the above equation) automatically implies its summation. For clarity, however, this convention will rarely be used in this text.