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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

      represents three linear forms:

      (1.2) image

      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 image.

      Solution: image

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

      Thus, aixi will represent a linear form

image

      For example, image 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.

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

      (1.3) image

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

      For example, image, by summation convention is expressed as

image

      We shall now consider some properties of this system.

      (1.4) image

      Property 1.4.2. From the summation convention, we get

image

      Similarly, δii = δii = n

      Property 1.4.3. From the definition of δi j, taken as an element of unit matrix I, we have

image

       Property 1.4.4.

      (1.6) image

      (1.7) image

image

      Also, by definition, image

      In particular, when i = k, we get image

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

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

      (1.8a) image

      (1.8b) image

      (b) image by 1.8b

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

image

      Since xj is independent of, image when j i

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