00072___1ff2002cabce87e09d6486f642544050 - “absolute”...

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Sec. 2.13 TENSOR EQUATIONS 51 As a second application we can prove the following theorem: the co- variant derivatives of the Euclidean metric tensor gij and the associated contravariant tensor gij are zero: (2) gijlk=o, g23Jk=o. .. Since gij Ik and gij Ik are tensors, the truth of the theorem can be established if we can demonstrate Eq. (2) in one particular coordinate system. But this is exactly the case in Cartesian coordinates, in which gij and gaj are constants and, hence, their derivative vanish. Thus, the proof is completed. In contrast to Eq. (l), however, it can be proved that Eqs. (2) remain true in Riemannian space. Further examples are furnished in Probs. 2.17 to 2.23. To apply this powerful procedure, one must make sure that all quantities involved are tensors. In particular, we must ascertain that all scalars are
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Unformatted text preview: “absolute” constants. This remark is very important because in physics we also use quantities that transform like relative tensors. A reZative tensor of weight w is an object with components whose transformation law differs from the tensor transformation law by the appearance of the Jacobian to the wth power as a factor. Thus, (3) (4) are the transformation laws for a relative scalar field of weight w and a relative contravariant vector field of weight w, respectively. If w = 0, we have the previous notion of a tensor field. Whether an object is a tensor or a relative tensor is often a matter of definition. As an example, consider the total mass enclosed in a volume expressed in terms of density. Let xi be rectangular coordinates which are transformed into curvilinear coordinates 0,. We have (5)...
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