differential geometry w notes from teacher_Part_27

differential geometry w notes from teacher_Part_27 - 53...

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Unformatted text preview: 53 2.4. TENSORS Theorem 2.4.1 Let n A= Ai dxi i =1 be a covector field (1-form). Let Fi j = ∂i A j − ∂ j Ai . • Then n F= Fi j dxi ⊗ dx j i, j=1 is a tensor of type (0, 2). Proof : 1. Check the transformation law. • A tensor is called isotropic if it is a tensor product of g, g−1 and I . • The components of an isotropic tensor are the products of gi j , gi j and δij . • Every isotropic tensor of type ( p, q) has an even rank p + q. • For example, the most general isotropic tensor of type (2, 2) has the form Ai j kl = agi j gkl + bδik δlj + cδil δkj , where a, b, c are scalars. 2.4.8 Einstein Summation Convention • In any expression there are two types of indices: free indices and repeated indices. • Free indices appear only once in an expression; they are assumed to take all possible values from 1 to n. • The position of all free indices in all terms in an equation must be the same. diffgeom.tex; April 12, 2006; 17:59; p. 56 54 CHAPTER 2. TENSORS • Repeated indices appear twice in an expression. It is assumed that there is a summation over each repeated pair of indices from 1 to n. The summation over a pair of repeated indices in an expression is called the contraction. • Repeated indices are dummy indices: they can be replaced by any other letter (not already used in the expression) without changing the meaning of the expression. • Indices cannot be repeated on the same level. That is, in a pair of repeated indices one index is in upper position and another is in the lower position. • There cannot be indices occuring three or more times in any expression. diffgeom.tex; April 12, 2006; 17:59; p. 57 ...
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