TABLE OF CONTENTS It is now convenient to briefly review the mathematics of tensors. Tensors are of primary importance in connection with coordinate transforms. They serve to isolate intrinsic geometric and physical properties from those that merely depend on coordinates.
A tensor of rank r in an n-dimensional space possesses n r components which are, in general, functions of position in that space. A tensor of rank zero has one component, A, and is called a scalar. A tensor of rank one has n components, (A 1, A 2, , A n), and is called a vector. A tensor of rank two has n 2 components, which can be exhibited in matrix format. Unfortunately, there is no convenient way of exhibiting a higher rank tensor. Consequently, tensors are usually represented by a typical component: e.g., the tensor A ijk (rank 3), or the tensor A ijkl (rank 4), etc. The suffixes i,j,k, are always understood to range from 1 to n.
For reasons which will become apparent later on, we shall represent tensor components using both superscripts and subscripts. Thus, a typical tensor might look like A ij (rank 2), or B i j (rank 2), etc. It is convenient to adopt the Einstein summation convention. Namely, if any suffix appears twice in a given term, once as a subscript and once as a superscript, a summation over that suffix (from...
