00062___9b7b5098b7c0bd9bd7df687a96fe4fa0

00062___9b7b5098b7c0bd9bd7df687a96fe4fa0 - = 1gijl > 0...

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Sec. 2.6 SOME IMPORTANT SPECIAL TENSORS 41 ranks 2 and 3 in rectangular Cartesian coordinates XI, 22, x3, =O wheni#j, = 1 (1) 6Zj = 6ij = when i = j, j not summed , = 0 = 1 = -1 when any two indices are equal, when i, j, Ic permute like 1,2,3, when i,j, Ic permute like 1,3,2, eijk = eijk (2) what will be their components in general coordinates 0'? The answer is provided immediately by the tensor transformation laws. Thus, ax" ax" -- ax" ax" a0i 803 (3) gij = asiasj6rnn = (4) (5) (7) where g is the value of the determinant 1gijl and is positive for any proper coordinate system, i.e., (8)
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Unformatted text preview: = 1gijl > 0 We see that the proper generalizations of the Kronecker delta are the Euclidean metric tensors, and those of the permutation symbol are f i j k and eijk, which are called permutation tensors or alternators. As defined by Eqs. (6) and (7), the components of Eijk and eijk do not have values 1, -1, or 0 in general coordinates, they are (a, -a, 0) and (l/fi, -l/fi, 0) respectively. Note that the mixed tensor gj is identical with 6; and is constant in all coordinate systems. Fkom Eqs. (3) and (4) we see that (9) gimgmj = 6: . Hence, the determinant .. IgimgmjI = 1gij1 ' 1ga31 = 16;1 = 1....
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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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