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Archived: 07 Aug 2014, 10:06
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deant
Joined: 12 Jul 2010, 07:44 Posts: 154
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 Exercise [19.17]
Suppose that: (*)  for every vector  . Let i, j be any two particular coordinate indices. (i.e. these are NOT abstract indices, they're constants, in the range 0..3 when working in four dimensions). Define the vectors: p = q = r =  (e.g. for i=1 and j=3, p = [0,1,0,0], q = [0,0,0,1], r = [0,1,0,1]; for i= j=0, p, q = [1,0,0,0], r = [2,0,0,0]; ...etc.) Then:    . Combining the above gives: (1)  And similarly: (2)  By (*), the RHS of (1) and of (2) must be equal; so we have proven that  for every choice of i, j. Hence  . If  and  are symmetric tensors,  and  , and so in that case (*) implies that  . 
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07 Apr 2012, 18:59 |
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deant
Joined: 12 Jul 2010, 07:44 Posts: 154
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 Re: Exercise [19.17]
Another way of looking at this is to note that any (arbitrary) symmetric ![\left[{}^2_0\right] \left[{}^2_0\right]](latexrender/pictures/caefd9c460547a5e7aaa28b1511c2fe0.png) tensor  can be constructed as a sum of vector (outer-)products:  So (*) implies that  for any  . This is equivalent to  for any  (even non-symmetric). ...So once again we get  .
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07 Apr 2012, 19:30 |
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