Mathematical Preliminaries

Abstract

Physical quantities and concepts of interest in continuum mechanics, for example, displacement, force, acceleration, strain, stress, strain-rate, etc., are defined independent of a coordinate system. Hence, fundamental to quantification in continuum mechanics are coordinate invariant mathematical constructs such as vectors and tensors. This chapter briefly introduces the requisite concepts of vector and tensor analysis, first in terms of a direct notation and then referenced to Cartesian and orthogonal curvilinear coordinates, and, because many computations are simplified by writing the components of vectors and tensors in matrix form, summarizes some results from matrix methods. An understanding of the ideas presented in this chapter is sufficient for embarking on the subsequent chapters in this text, but the interested reader is encouraged to consult more complete sources: Bowen and Wang (1976), Simmonds (1982), Golub and van Loan (1983), and Knowles (1998) as well as selected chapters in Malvern (1969), Chadwick (1976), Gurtin (1981), and Antman (1995).