Essentials of Mathematical Methods in Science and Engineering, 2nd Edition

CHAPTER 3GENERALIZED COORDINATES AND TENSORS

Scalars are quantities that can be defined by just giving a single number. In other words, they have only magnitude. Vectors on the other hand, are defined as directed line segments, which have both magnitude and direction. By assigning a vector to each point in space, we define a vector field. Similarly, a scalar field can be defined as a function of position. Field is one of the most fundamental concepts of theoretical physics. In working with scalars and vectors, it is important that we first choose a suitable coordinate system. A proper choice of coordinates that reflects the symmetries of the physical system, simplifies the algebra and the interpretation of the solution significantly. In this chapter, we start with the Cartesian coordinates and their transformation properties. We then show how a generalized coordinate system can be constructed from the basic principles and discuss general coordinate transformations. Definition of scalars and vectors with respect to their transformation properties brings new depths into their discussion and allows us to introduce more sophisticated objects called tensors. We finally conclude with a detailed discussion of cylindrical and spherical coordinate systems, which are among the most frequently encountered coordinate systems in applications.

3.1 TRANSFORMATIONS BETWEEN CARTESIAN COORDINATES

Transformations between Cartesian coordinates that exclude scale changes:

(3.1)

are called