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X , is usually denoted as xi,i = 1 , .. ,n. A set of n variables y ' , y 2 , . . , y n is denoted by y i , i = 1 , .. ,n. We emphasize that y l , y 2 , . . , yn are n independent variables and not the first n powers of the variable y . Consider an equation describing a plane in a three-dimensional space X I , x2,x3, (1) alxl+ a2z2 + a3x3 = p , where ai and p are constants. This equation can be written as 3 i=l However, we shall introduce the summation convention and write the equation above in the simple form The convention is as follows: The repetition of a n index (whether superscript or subscript) in a term will denote a summation with respect to that index over its range.
40 Chap. 2 TENSOR ANALYSIS Mixed tensor field of rank two, t3: (3) Generalization t o tensor fields of higher ranks is immediate. p, a tensor field of rank r = p + q, contravariant of rank p and covariant of rank q , if the components in any two coordinate systems are related by Thus, the location of an index is important in telling whether it is contravariant or covariant. Again, if only rectangular Cartesian coordinates are considered, the distinction disappears. These definitions can be generalized in an obvious manner if the range of the indices are 1 , 2 , .
The fact that the length of the vector is unity is expressed by the equation (ad2 (a2I2 (w3)2= 1 , + + or simply aiai = 1 . (4) As a further illustration, consider a line element ( d x , d y , d z ) in a threedimensional Euclidean space with rectangular Cartesian coordinates x , y , z . The square of the length of the line element is (5) ds2 = d x 2 + d y 2 + d z 2 . If we define (6) dx’ = d x , d x 2 = d y , dx3 = d z , and Then (5) may be written as (8) ds2 = bijdxidx’, with the understanding that the range of the indices i and j is 1 to 3.