Applied Mathematics for Science and Engineering

3Vectors and Tensors

Introduction

A significant area of interest in continuum mechanics is the application of forces to material objects and the response of those objects to the applied forces. Most of us tend to think of a force simply as a directed line segment (possessing magnitude and direction), but a little caution is in order here: We must remember that the effect of a force applied to a body depends on both its line of action and the point at which it is applied. For example, if one were using a lever to move a boulder, the distance between the point of application and the fulcrum would have enormous impact on the effectiveness of the action. We also need to emphasize that our discussions here are concerned with continuum mechanics in three-dimensional Euclidean space. Thus, when we speak of tensors, for example, we mean Cartesian tensors. Tensors do figure prominently in non-Euclidean spaces, but those applications are not relevant to our principal objectives.

We will begin by reviewing what we mean when we refer to scalars, vectors, and tensors. A scalar is a quantity that has magnitude only; for example, we might find that an enclosure has a volume of 1.2 m³ or that the fluid contained within has a temperature of 215°F (101.67°C). We also observe that a scalar is a zero-order tensor (we will use order and rank synonymously). In contrast, a vector has both magnitude and direction, and we can think of force and velocity as examples. A vector with three components is a first-order ...