Advanced Mathematics For Engineering and Science
As the product of a close collaboration between two mathematicians and an engineer, this pragmatic text provides the know-how and know-why to understanding mathematics and its many applications.
TABLE OF CONTENTS Advanced Mathematics For Engineering and Science
Chapter 4: Vector and Tensor Analysis
4.1 INTRODUCTION
In this chapter, we will be dealing mainly with vectors and second order tensors. A vector is defined as a quantity that has both magnitude and direction. In a three-dimensional space, it is described by three numbers (components). For example, velocity is a vector and, when referred to rectangular Cartesian axes, it is specified by its three components (v x, v y, v z). A vector is also defined as a tensor of order one. A scalar has only magnitude and is completely characterized by one number. It is a tensor of order zero and temperature is an example of a scalar. A second order tensor, in a three-dimensional space, is represented by nine numbers (components). The extra stress tensor in fluid mechanics is an example of a second order tensor and can be denoted by ? xx, ? xy, ? xz, ? yx, ? yy, ? yz, ? zx, ? zy, and ? zz.. Third and fourth order tensors will also be introduced.
4.2 VECTORS
In this section, we briefly review some of the known and useful results of vector algebra and of vector calculus. In Table 4.2 1, v, a and q are vectors, c 1 and c 2 are scalars and ? and ? are differentiable scalar functions of position x, y, z, or time t. i, j and k
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