00067___399066c80426672334fafa139530bf1b - 46 Chap. 2 T...

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46 TENSOR ANALYSIS Chap. 2 The proof is very simple. Since A(a,j, k)Ea is of type A;, it is trans- formed into %coordinates as But <p = (axfl/aP)<*. Inserting this in the right-hand side of the equation above and transposing all terms to one side of the equation, we obtain Now is an arbitrary vector. Hence, the bracket must vanish and we have which is precisely the law of transformation of the tensor of the type A&. The pattern of the example above can be generalized to prove the theo- rem that, if [A(il, i2,. . . , i')] is a set of functions of the variables xi, and if the product A(a, . . , iT)E" with an arbitrary vector E" be a tensor of the type A, ,,,. kp(x), where p + q = r, then the set A(i1, . . , i,.) represents a tensor of the type A$~;'~~,p(x). Similarly, if the product of a set of n2 functions A(a, j) with an arbitrary tensor B,k (and is summed over a) is a covariant tensor of rank
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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