00051___e13fd7919311ea6b77f5fc7cca9807ce

00051___e13fd7919311ea6b77f5fc7cca9807ce - where ai and p...

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2 TENSOR ANALYSIS In attempting to develop the theories outlined in the previous chapter rigorously and succinctly, we shall first learn to use the powerful tool of tensor calculus. At first sight, the mathematical analysis may seem in- volved, but a little study will soon reveal its simplicity. 2.1. NOTATION AND SUMMATION CONVENTION Let us begin with the matter of notation. In tensor analysis one makes extensive use of indices. A set of n variables 21, x2, . . . , x, is usually denoted as xi, i = 1,. . . ,n. A set of n variables y', y2,. . . ,yn is denoted by yi, i = 1,. . . , n. We emphasize that yl, y2,. . . , 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 XI, x2,x3,
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Unformatted text preview: where ai and p are constants. This equation can be written as alxl+ a2z2 + a3x3 = p , 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 an index (whether superscript or subscript) in a term will denote a summation with respect to that index over its range. The range of an index i is the set of n integer values 1 to n. A lower index i, as in ai, is called a subscript, and upper index i, as in xi, is called a superscript. An index that is summed over is called a dummy index; one that is not summed out is called a free index. Since a dummy index just indicates summation, it is immaterial which symbol is used. Thus, aixa may be replaced by ajxj, etc. This is analogous 30...
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