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Easy tensor question

  • Thread starter Ibix
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  • #1
Ibix
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Another trivial question from me.

Homework Statement



Which (if any) of the following are valid tensor expressions:
(a)[itex]A^\alpha+B_\alpha[/itex]
(b)[itex]R^\alpha{}_\beta A^\beta+B^\alpha=0[/itex]
(c)[itex]R_{\alpha\beta}=T_\gamma[/itex]
(d)[itex]A_{\alpha\beta}=B_{\beta\alpha}[/itex]

Homework Equations



Nothing relevant - these are generic tensors.

The Attempt at a Solution



(a) and (c) are not valid because the indices don't match up. (d) is valid - in matrix notation, A=BT.

I'm not sure about (b), though. The left hand side is valid; summing over the dummy index makes it a sum of two vectors. I'm not quite sure how to interpret the equality, though. I can see it as a vector equaling a scalar - which is not valid. Alternatively, I can read an implicit [itex]\forall \alpha[/itex] - in other words, that each element of the vector on the left hand side is identically zero.

I lean towards the first interpretation - but I'm not sure.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
TSny
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For an equation like (b), the zero on the right would normally be interpreted as the zero vector (rather than the zero scalar); or more precisely, it would be the ##\alpha## component of the zero vector (which would of course have a value of 0 in any coordinate system and any reference frame). So, I think your second interpretation is better.
 
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  • #3
Ibix
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Thank you very much.

This is self-study. When I downloaded the problem sheet the answers were available - now term has started and they've gone... Shouldn't have been conscientious.
 

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