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Exercise [19.17]
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Joined: 12 Jul 2010, 07:44
Posts: 154
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 . 07 Apr 2012, 18:59 Joined: 12 Jul 2010, 07:44
Posts: 154
Another way of looking at this is to note that any (arbitrary) symmetric 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 .

07 Apr 2012, 19:30 Page 1 of 1 [ 2 posts ] 