Continuum mechanics seeks to provide a fundamental model for material response. It is sensible to require that the predictions of such a theory should not depend on the irrelevant details of a particular coordinate system. The key is to write the theory in terms of variables that are unaffected by such changes; tensors (or tensor fields) are the measures that have this property. Tensors come in different flavors depending on the number of spatial directions that they couple. The simplest tensor has no directional dependence and is called a scalar invariant to distinguish it from a simple scalar. A vector has one direction. For two directions and higher the general term tensor is used.

Tensors are tricky things to define. Many books define tensors in a technical manner in terms of the rules that tensor components must satisfy under coordinate system transformations. While certainly correct, we find such definitions unilluminating when trying to answer the basic question of “what is a tensor?”. In this chapter, we provide an introduction to tensors from the perspective of linear algebra. This approach may appear rather mathematical at first, but in the end it provides a far deeper insight into the nature of tensors.

Before we can begin the discussion of the definition of tensors, we must start by defining “space” and “time” and the related concept of a “frame of reference,” which underlie the description of all physical objects.