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Chapter 5:
Tensor Analysis
Abstract
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. 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 3 n 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 3 n 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.