Introduction to Differential Geometry with Tensor Applications. Группа авторов

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).