00085___bb1d0ab1b5355eb6ec7aaee954fbf604 - 64 C hap 2...

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64 TENSOR ANALYSIS Chap. 2 Ans. = -xl, = l/zl, all other components = 0. 2.26. 2.27. Show that r&, is symmetric in m and n; i.e., rLn = rKm. Show that the necessary and sufficient condition that a given curvi- linear coordinate system be orthogonal is that gij = 0, if i # j, throughout the domain. 2.28. Prove that g, x g, = e,.Stgt, g' x g" = EPStgt, where erst, erst are the permutation tensor of Sec. 2.6. Hence, if we denote the scalar product of the vectors gl and gz x 93 by [glgags] or (g1,gz x g3), we have 123 [g1g2g31 = (g1,gz x g3) = &, [g g g 1 = (g',gZ x 2) = I/&. 2.29. The element of area of a parallelogram with two adjacent edges ds2 = g2d6' and ds3 = g3de3 is dS1 = Ids2 x dS3) = )gz x g31d62d63. Show that dS1 = dmd62d63. In general, the element of area dSi of a par- allelogram formed by the elements g,d@ and gkdek, (j, k not summed), on the &-surface is dS, = mde3dBk (i not summed, i # j # Ic). 2.30.
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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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