00077___350e281261e31ef2e67e43eed456a8d9 - g's and g's...

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56 TENSOR ANALYSIS Chap. 2 (m, r not summed). In general, pL # phm # p; # By. Here the first indices of the superscript and the subscript are associated with the coordi- nates with base vectors g's and the second indices with g's. The distinction between p; and prm disappear in Cartesian coordinates. One can easily show that (15) (16) gr = (gr . gm)gm = Phmgm , gm = (gm * gk)gk = (gk gm)gk = P(e,gk and that the p's are related by the metric tensors of the two coordinate systems: D; = Bmp& gr4 , P& = ii ''PAq gqm . From Eqs. (6) and (14), it can be seen that Thus, Eqs. (15) and (16) establish that the transformation between
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Unformatted text preview: g's and g's follows the tensor transformation law. A substitution of Eq. (16) into Eq. (15) and vice versa yield g~ = pik& = Pikp?grn gm = Pkmgk = P(e,&,'gr , which implies that We can now determine the transformation law for the components of v between different coordinate systems. Dotting both sides of the equation -r- 21 gr = vmgm with gi gives the transformation law (19) A vi = (gi ' g m ) P = p:mu". Similarly, dotting the same equation with gj gives the inverse transforma- tion Using the relations of the contravariant base vectors (21) A g' r= (g" . gm)gm = p&gm, gm = (g, . gm)g' = p;"gT,...
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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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