00061___a96454c39c20e9df52385c1dfb65fa64

00061___a96454c39c20e9df52385c1dfb65fa64 - 2.8. Prove the...

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40 TENSOR ANALYSIS Chap. 2 Mixed tensor field of rank two, t3: (3) Generalization to tensor fields of higher ranks is immediate. Thus, we call a quantity tp, ...p, a tensor field of rank r = p + q, contravariant of rank p and covariant of rank q, if the components in any two coordinate systems are related by al"'ap Thus, the location of an index is important in telling whether it is contravariant or covariant. Again, if only rectangular Cartesian coordinates are considered, the distinction disappears. These definitions can be generalized in an obvious manner if the range of the indices are 1,2,. . . , n. PROBLEMS 2.7. Show that, if all components of a tensor vanish in one coordinate system, then they vanish in all other coordinate systems which are in one-to-one correspondence with the given system. This is perhaps the most important property of tensor fields.
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Unformatted text preview: 2.8. Prove the theorem: The sum or difference of two tensors of the same type and rank (with the same number of covariant and the same number of con- travariant indices) is again a tensor of the same type and rank. Thus, any linear combination of tensors of the same type and rank is again a tensor of the same type and rank. 2.9. Theorem. Let A{:','.'&r, l3Ei:::kp be tensors. The equation is a tensor equation; i.e., if this equation is true in some coordinate system, then it is true in all coordinate systems which are in one-to-one correspondence with each other. Hint: Use the results of the previous problems. 2.6. SOME IMPORTANT SPECIAL TENSORS If we define the Kronecker delta and the permutation symbol introduced in Sec. 2.1 as components of covariant, contravariant, and mixed tensors of...
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