Notation and Systems of Numbers

2.1 Introduction and Basic Concepts

In order to get acquainted with the basic notation and concepts of the tensor calculus, it is convenient to use some well-known concepts from linear algebra. The collection of N elements of a column matrix is often denoted by subscripts as x1,x2,…,xN. Using a lower index i = 1,2,…,N, we can introduce the following short-hand notation:

i(i=1,2,…,N).

Sometimes, the same collection of N elements is denoted by corresponding superscripts as x1,x2,…,xN. Using here an upper index i = 1,2,…,N, we can also introduce the following short-hand notation:

i(i=1,2,…,N).

In general the choice of a lower or an upper index to denote the collection of N elements of a column matrix is fully arbitrary. However, it will be shown later that in the tensor calculus lower and upper indices are used to denote mathematical objects of different natures. Therefore, both types of indices are essential for the development of tensor calculus as a mathematical discipline. In the definition (2.2) it should be noted that i is an upper index and not a power of x. Whenever there is a risk of confusion of an upper index and a power, such as when we want to write a square of xi, we will use parentheses as follows:

i⋅xi=(xi)2(i=1,2,…,N).

A collection of numbers, defined by just one (upper or lower) index, will be called a first-order system or a simple system. The individual elements of such a system will be called the elements or coordinates of the system. The introduction of the lower and upper indices provides a device to highlight the different nature of different first-order systems with the equal number of elements. Consider, for example, the following linear form:

x+by+cz.

Introducing the labels ai = {a,b,c} and xi = {x,y,z}, the expression (2.4) can be written as follows:

1x1+a2x2+a3x3=∑i=13aixi

indicating the different nature of the two first-order systems. In order to emphasize the advantage of the proposed notation, let us consider a bilinear form created using two first-order systems xi and yi(i = 1,2,3):

a11x1y1+a12x1y2+a13x1y3+a21x2y1+a22x2y2+a23x2y3+a31x3y1+a32x3y2+a33x3y3=∑i=13∑j=13aijxiyj.

Here, we see that the short-hand notation on the right-hand side of (2.6) is quite compact. The system of parameters of the bilinear form

ij(i,j=1,2,3)

is labeled by two lower indices. This system has nine elements and they can be represented by the following 3 × 3 square matrix:

11a12a13a21a22a23a31a32a33.

A system of quantities determined by two indices is called a second-order system.

Depending on whether the indices of a second-order system are upper or lower, there are three types of second-order systems:

ij,aij,aij(i,j=1,2,…,N).

A second-order system in N dimensions has N2 elements. In a similar way, we can define the third-order systems, which may be of one of the following four different types:

ijk,ajki,akij,aijk(i,j=1,2,…,N).

The most general system of order K is denoted by

i1,i2,…,iK(i1,i2,…,iK=1,2,…,N)

and depending on the position of the indices, it may be of one of several different types. The Kth-order system in N dimensions has NK elements.

2.2 Symmetric and Antisymmetric Systems

Let us consider a second-order system in three dimensions

ij(i,j=1,2,3).

The system (2.12) is called a symmetric system with respect to the two lower indices if the elements of the system satisfy the equality

ij=aji(i,j=1,2,3).

Similarly, the system (2.12) is called an antisymmetric system with respect to the two lower indices if the elements of the system satisfy the equality

ij=−aji(i,j=1,2,3).

The equality (2.14) indicates that an antisymmetric second-order system in three dimensions has only three independent components and that all the diagonal elements are equal to zero:

JJ=0(J=1,2,3).

Thus, it is possible to represent an antisymmetric second-order system in three dimensions by the following 3 × 3 matrix:

a12a13−a120a23−a13−a230.

In general, a system of an arbitrary order and type will be symmetric with respect to two of its indices (both upper or both lower), if the corresponding elements remain unchanged upon interchange of these two indices. The system will be totally symmetric with respect to all upper (lower) indices, if an interchange of any two upper (lower) indices leaves the corresponding system elements unchanged. Elements of a totally symmetric third-order system with all three lower indices satisfy the equality

ijk=aikj=ajki=ajik=akij=akji.

Analogous to the above, a system of an arbitrary order and type will be antisymmetric with respect to the two of the indices (both upper or both lower), if the corresponding elements change signs upon interchange of these two indices. The system will be totally antisymmetric with respect to all upper (lower) indices, if an interchange of any two upper (lower) indices changes signs of the corresponding system elements. Elements of a totally antisymmetric third-order system with all three lower indices satisfy the equality

ijk=−aikj=ajki=−ajik=akij=−akji.

2.3 Operations with Systems

Under certain conditions, it is possible to perform a number of algebraic operations with systems. The definition of these operations depends on the order and type of the systems.

2.3.1 Addition and Subtraction of Systems

The addition and subtraction of systems can be performed only with the systems of the same order and same type. The addition (subtraction) of systems is performed in such a way that each element of one system is added (subtracted) to (from) the corresponding element of the other system (the one with the same indices in the same order). For example, the systems kmij and kmij can be added since they are of the same order and of the same type. The sum of these two systems is given by

kmij=Akmij+Bkmij,

and it is the system of the same order and type as the two original systems. This definition is easily extended to addition and subtraction of an arbitrary number of systems.

2.3.2 Direct Product of Systems

A system obtained by multiplying each element of one system by each element of another system, regardless of their order and type, is called a direct product or just a product of these two systems. Thus, for example, a product of two first-order systems ai and bi is a second-order system

ij=aibj.

For i,j = 1,2,3 this operation can be written in the following matrix form:...