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Can tensor products define (M,N) tensors?

  1. Jun 13, 2014 #1
    From my understanding, an arbitrary (0,N) tensor can be expressed in terms of its components and the tensor products of N basis one-forms. Similarly, an arbitrary (M,0) tensor can be expressed in terms of its components and tensor products of its M basis vectors. What about an (M,N) tensor, though? I'm reading from Schutz right now and he doesn't mention anything like that, but it seems pretty logical to me that you should be able to have a tensor product between basis vectors and basis one-forms. If you did that then every (M,N) tensor could be expressed just in terms of its components and the M basis vectors and N basis one-forms. The only thing you'd need to keep track of is the order in which you 'supply' the (M,N) tensor with vectors and one-forms (since obviously you can't support a basis one-form with a one-form, and likewise for basis vectors), but you have to do that anyway for (0,N) and (M,0) tensors as well since the tensor product isn't commutative.

    Is this type of product defined? Is it the same as the one between one-forms and one-forms, and between vectors and vectors?

    Thanks!
     
  2. jcsd
  3. Jun 13, 2014 #2

    Matterwave

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    Certainly:

    $$T=\sum_{ijk...lmn...}T_{ijk...}^{lmn...}\tilde{q}^i\otimes\tilde{q}^j\otimes\tilde{q}^k\otimes...\otimes \vec{e}_l\otimes\vec{e}_m\otimes\vec{e}_n\otimes...$$
     
  4. Jun 14, 2014 #3
    Ah cool thank you! It seemed reasonable but I just wanted to make sure because I'd never seen it stated explicitly.
     
  5. Jun 14, 2014 #4

    Fredrik

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    I see that the question has been answered, but I think an example of what you can do may be useful:
    \begin{align}
    &T(\omega,u,v)=T(\omega_i e^i,u^j e_j,v^k e_k)=\omega_i u^j v^k T(e^i,e_j,e_k) = e_i(\omega)e^j(u)e^k(v) T^i{}_{jk} = (T^i{}_{jk} e_i\otimes e^j\otimes e^k)(\omega,u,v)\\
    &T=T^i{}_{jk} e_i\otimes e^j\otimes e^k
    \end{align}
     
  6. Jun 14, 2014 #5
    Thanks for the example :)
     
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