Preliminary Mathematics

Abstract

This chapter addresses mathematical preliminaries necessary to understand polymer viscoelasticity assuming that the readers are familiar with engineering mathematics of sophomore. Analysis of vector and tensor is the majority of this chapter, which is necessary to understand constitutive theories of polymer viscoelasticity as well as the theory of polymer physics. Since the knowledge of functional analysis is also needed to understand numerical methods to be used for the processing of viscoelastic data, the vectors and tensors in this chapter include not only physical quantities but also generalized ones called abstract vectors. Because of this purpose, the analysis of vector and tensor starts from the notion of vector space which is an abstraction of physical vector. As for linear viscoelastic theory, both Fourier and Laplace transforms are frequently used. Since this book is not a text of mathematics, rigorous proofs will not be seriously considered. For the proofs, the readers should refer the related references.