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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 (1form). 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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 Spring '10
 Wong
 Geometry

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