Scalars, vectors, and Cartesian tensors

Geometry is a vital ingredient in the description of continuum problems. Our treatment will focus on the mathematically simplest representation for this subject. Although curvilinear coordinates can be more natural, they introduce complications that go beyond the scope of this book. The initial part of our treatment will parallel the Cartesian approach of Mase and Mase (1990) rather than the curvilinear approach of Narasimhan (1993) and Fung (1965). Hence, we will adhere to a Cartesian description of problems and be spared the need to distinguish between covariant and contravariant notation. Moreover, we will generally employ second-rank tensors which are matrices that possess some very special and important (coordinate) transformation properties.

We will distinguish between three classes of objects: namely, scalars, vectors, and tensors. In reality, all quantities may be regarded as tensors of a specific rank. Scalar (nonconstant) quantities, such as density and temperature, are zero rank or order tensors, while vector quantities (which have an associated direction, such as velocity) are first-rank tensors. Second-rank tensors, such as the stress tensor, are a special case of square matrices. We will usually denote vector quantities by bold-face lower-case letters, while second-rank tensors will be denoted by bold-face upper-case letters.