00076___2d61608f6d02a4866921bc7f1d63f4a6

00076___2d61608f6d02a4866921bc7f1d63f4a6 - Sec. 2.14 G...

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Sec. 2.14 GEOMETRIC INTERPRETATION OF TENSOR COMPONENTS 55 where grs is the Euclidean metric tensor of the coordinate system, and grS is the associated, or conjugate metric tensor. From Eq. (9) it is clear that the contravariant base vectors g1 , g2, g3 are, respectively, perpendicular to the planes of g2g3,g3glrg1g2. (See also Prob. 2.28.) For orthogonal coordinates, that means gi - gj = 0 for i # j, it can be shown that is in the same direction as and that . = 0 i # j. We can now deal with the contravariant and covariant components of a vector. Consider the expression v'g,., where wr is a contravariant tensor of rank one, and g,. are the covariant base vectors at a generic point. This expression remains invariant under coordinate transformations, and, since it is a linear combination of the base vectors g,., it is a vector, which may be designated v: By Eq. (lo), replacing g, by grSgs, we also have (11) v = vrgr * where (13) A v, = grswr, vr = grsvs. According to Eq.
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