chapter 5, Tensor Analysis

Excerpt

In this chapter we introduce the basic transformation laws of tensors that are used in various engineering application areas like elasticity and general relativity, among others. Because many of the tensors of interest are of second order, we find the matrix operations introduced in Chapter 3 to be particularly useful in our treatment here.

5.1 Introduction

A scalar is a quantity that can be specified (in any coordinate system) by just one number, whereas the specification of a vector requires three numbers (see Chapter 4). Both scalars and vectors are special cases of a more general concept called a tensor. To specify a tensor of order n in a coordinate system requires 3ⁿ numbers, called the components of the tensor. Scalars are tensors of order 0 and vectors are tensors of order 1.

A tensor of order n is more than just a set of 3ⁿ numbers. Only when these numbers satisfy a particular transformation law do they represent the components of a tensor. The transformation law describes how the tensor components in one coordinate system are related to those in another coordinate system. Because they have useful properties that are independent of coordinate system, tensors are used to represent various fundamental laws of physics, engineering, science, and mathematics. In particular, tensors are an important tool in general relativity, elasticity, hydrodynamics, and electromagnetic theory. In these areas of application the elastic, optical, electrical, and magnetic properties must often be described by tensor quantities. For example, this is the case if the medium is anisotropic, like in many crystals, or is a plasma in the presence of a magnetic field.

As a final comment, we alert the reader to the change in the use of certain punctuation marks in the present chapter to avoid confusion with the “comma” notation commonly used for partial derivative in this material. For example, when deemed necessary, a semicolon (;) is often used in place of a comma.

5.2 Tensor Notation

The use of vector symbols like a and a + b to denote a vector or vector operation provides a convenient notation for expressing relationships in geometry and physics (see Chapter 4). For actually performing the various operations between vectors, such as addition and multiplication, it is the components of the vector that are most useful. Shorthand notation based on vector components that is widely used in advanced works is that involving tensor. We briefly introduced tensor notation for Cartesian coordinates in Chapter 4, but now we wish to build upon that notation for more general coordinate systems that require both upper and lower indices.

In tensor notation we can represent a triad of numbers (x₁,x₂,x₃) by the symbol x_j, where j = 1, 2, or 3. We refer to the subscript j in this context as a dummy index. In particular, if a vector a can be represented in component form a − 〈a₁,a₂,a₃〉, then by use of the index notation we may also represent these components by