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1
Preliminaries

1.1 Introduction

Some quantities are associated with their magnitude and direction, but certain quantities are associated with two or more directions. Such a quantity is called a tensor, e.g., the stress at a point of an elastic solid is an example of a tensor which depends on two directions: one is normal and the other is that of force on the area. Tensor comes from the word tension.

In this chapter, we discuss the notation of systems of different orders, which are applied in the theory of determinants, symbols, and summation conventions. Also, results on some matrices and determinants are discussed because they will be used frequently later on.

1.2 Systems of Different Orders

Let us consider the two quantities, a₁, a₁ or a¹, a², which are represented by ai or ai, respectively, for i = 1, 2. In such cases, the expressions ai, ai, ai j, ai j, and are called systems. In each value of ai and ai are called systems of first order and each value of ai j, ai j, and is called a double system or system of second order, of which a₁₂, a₂₂a²³, a¹³, and are called their respective components. Similarly, we have systems of the third order that depend on three indices shown as ai jk, aikl, ai jm, ai jn, and and each number of their respective components are 8.

In a system of order zero, it is shown that the quantity has no index, such as a. The upper and lower indices of a system are called its indices of contravariance and covariance, respectively. For a system of , i and j are indices of a contravariant and k is of covariance. Accordingly, the system Aij is called a contravariant system, Aklm is called a covariant system, and is called a mixed system.

1.3 Summation Convention Certain Index

If in some expressions a certain index occurs twice, this means that this expression is summed with respect to that index for all admissible values of the index.

Thus, the linear form has an index, i, occurring in it twice. We will omit the summation symbol Σ and write aixi to mean a₁x₁ + a₂x₂ + a₃x₃ + a₄x₄. In order to avoid Σ, we shall make use of a convention used by A. Einstein which is accordingly called the Einstein Summation Convention or Summation Convention.

Of course, the range of admissible values of the index, 1 to 4 in this case, must be specified. If the symbol i has a range of values from 1 to 3 and j ranges from 1 to 4, the expression

(1.1)

represents three linear forms:

(1.2)

Here, index i is the identifying (free) index and since index j, occurs twice, it is the summation index.

We shall adopt this convention throughout the chapters and take the sum whenever a letter appears in a term once in a subscript and once in superscript or if the same two indices are in subscript or are in superscript.

Example 1.3.1. Express the sum .

Solution:

1.3.1 Dummy Index

The summation (or dummy) index can be changed at will. Thus, Equation (1.1) can be written in the form a_ikx_k if k has the same range of values as j.

We will assume that the summation and identifying indices have ranges of value from 1 to n.

Thus, aixi will represent a linear form

For example, can be written as aikxixk and here, i and k both are dummy indexes.

So, any dummy index can be replaced by any other index with a range of the same numbers.

1.3.2 Free Index

If in an expression an index is not a dummy, i.e., it is not repeated twice, then it is called a free index. For example, for ai jxj, the index j is dummy, but index i is free.

1.4 Kronecker Symbols

A particular system of second order denoted by , is defined as

(1.3)

Such a system is called a Kronecker symbol or Kronecker delta.

For example, , by summation convention is expressed as

We shall now consider some properties of this system.

Property 1.4.1. If x¹, x², … xn are independent variables, then

(1.4)

Property 1.4.2. From the summation convention, we get

Similarly, δ_ii = δ_ii = n

Property 1.4.3. From the definition of δ^{i j}, taken as an element of unit matrix I, we have

Property 1.4.4.

(1.5)

(1.6)

(1.7)

Property 1.4.5.

Also, by definition,

In particular, when i = k, we get

Remark 1.4.1. If we multiply xk by , we simply replace index k of xk with index i and for this reason, is called a substitution factor.

Example 1.4.1. Evaluate (a) and (b) where the indices take all values from 1 to n.

(1.8a)

(1.8b)

(b) by 1.8b

Example 1.4.2. If xi and yi are independent coordinates of a point, it is shown that

Solution: The partial derivative of ϕ in two coordinate systems are different and are connected by the following formula of Differential Calculus:

(1.9a)

Since xj is independent of, when j ≠ i

(1.9b)

Hence, the result follows from (1.9a) and (1.9b).

1.5 Linear Equations

Let us consider n linear equations such that

(1.10a)

where x¹, x², …. xn are n unknown variables.

Let us consider:

For the expansion of det |ai j| in terms of cofactors we have

(1.10b)

where a = |ai j| and the cofactor of ai j is Ai j.

We can derive Cramer’s Rule for the solution of the system of n linear equations:

Now, multiplying both sides of (1.10a) by Ai j, we get

by (1.10b), we get, axj = biAi j.

From here, we can easily get

Example 1.5.1. Show that , where a is a determinant ai jie a = |ai j| of order 3 and Ai j are cofactors of ai j.

Solution: By expansion of determinants, we have:

Which can be written as a_1jA^1j = a a_1jA^2j = 0 and a_1jA^3j = 0 [we know aijAij = a].

Similarly, we have

Using Kronecker Delta Notation, these can be combined into a single equation:

All nine of these equations can be combined into .

1.6 Results on Matrices and Determinants of Systems

It is known that if the range of the indices of a system of second order are from 1 to n, the number of components is n². Systems of second order are organized into three types: ai j, ai j, and their matrices,

each of which is an n × n matrix.

We shall now establish the following results:

Property 1.6.1. If , then and .

Proof: We shall prove this result by taking the range of the indices from 1 to 2, but the results hold, in general, when they range from 1 to n.

We get . Hence, .

Taking the determinant of both sides, we get , as we know |AB| = |A||B|.

Property 1.6.2. If , then, and , where (bik)^T is the transpose of

Proof: We have , hence, .

Therefore,

Taking determinants of both sides, we get (since │AT│ = │A│).

Property 1.6.3. Let the cofactor of the element in the determinant be denoted by . Then, by summation convention we have

If the cofactor of aij is represented by Akj, it is expressed by the equation:

If we divide the cofactor Akj of the element of akj by the value a of the determinant, we form the normalized cofactor, represented by:

The above equation becomes

Property 1.6.4. Let us consider a system of n linear equations:

for n unknown xi, where

, where is cofactor of .

, which is called Cramer’s Rule, for the solution of n linear equations.

Property 1.6.5. Considering the transformation zi = zi(yk) and yi = yi(xk), let N function zi(yk) be of independent N variables of yk so that .

Here, N equation zi = zi(yk) is solvable for the z’s in terms terms of yi’s.

Similarly, yi = yi(xk) is a solution of yi in terms of xi’s so that .

Now, we have by the chain rule of differentiation that

Taking the determinant, we get

(1.11)

Considering a particular case in which zi = xi, Equation (1.5) becomes

Or

This implies that the Jacobian of Direct Transformation is the reciprocal of the Jacobian of Inverse Transformation.

1.7 Differentiation of a Determinant

Consider the determinant and let the element be a function of x₁, x₂ … xn, etc. Let be the cofactor of of det a.

Then, the derivative of a with respect to x₁ is given by

Therefore, in general, we can write .

1.8 Examples

Example 1.8.1. Write the terms contained in S = aijxixj taking n = 3.

Solution: Since the index i (or j) occurs both in subscript and superscript, we first sum on i from 1 to 3, then on j from 1 to 3.

Example 1.8.2. Express the sum of .

Solution: Here, the number of terms is 3³ = 27.

Since the index i (or j or k) occurs both in subscript and superscript, we first sum on i from 1 to 3, then on each term of its 3 terms we sum j from 1 to 3. This results in 9 terms. Then, on each of the 9 terms we sum k from 1 to 3, which results in 27 terms. Like the last example, we sum

Example 1.8.3. If f is a function of n variables xi, write the differential of f.

Solution: Since f = f(x¹, x², … xn),

from calculus, we have

Example 1.8.4. (a) If apqxpxq = 0 for all values of the independent variables x¹, x², … xn and apq‘s are constant, show that aij + aji = 0.

(b) If apqrxpxqxr = 0 for all values of the independent variables x¹, x², … xn and apqr‘s are constant, show that akij + akji + aikj + ajki + aijk + ajik = 0.

Solution: Differentiating:

(1.12a)

with respect to xi

(1.12b)

Differentiating (1.12b), with respect to xj, we get

(b) Differentiating

with respect to xi

Differentiating with respect to xj, we get

Differentiating in the same way, with respect to xk we get

Example 1.8.5. If is a double system such that , show that .

Solution: We have , taking determinant ,

Example 1.8.6. If is a double system such that , show that either or .

Solution: From above result

Example 1.8.7. If and , show that

(1.13a)

(1.13b)

The above result can be stated as . It is the result of the multiplication of two determinants of the third order.

1.9 Exercises

1 1. Write out in full the following expression.

2 2. Expand the following using the summation convention.

3 3. Prove the following.

4 4. Show that for all values of independentvariables, x1, x2, … .xn, and where xp’s are constants.

5 5. Calculate

6 6. Using the relation , show that

7 7. Express each of the following sums using the summation convention:

8 8. Evaluate each of the following (range of indices 1 to n):9.

9 10. If yi are n independent functions of variables xi and zi are n independent functions of yi and if and then show that .

10 11. If and a−1 times the cofactor of in the determinant of show that

11 12. Prove that where