Einstein index notation

  • #1
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Main Question or Discussion Point

Hello
I am doing some exercises in continuum mechanics and it is a little bit confusing. I am given the following equations ## A_{ij}= \delta_{ij} +au_{i}v_{j} ## and ## (A_{ij})^{-1} = \delta_{ij} - \frac{au_{i}v_{j}}{1-au_{k}v_{k}}##. If I want to take the product to verify that they give the identity matrix (its components maybe is more accurate), should I change in one of the expressions the index letters and proceed(change the free indices I mean)? Is this the correct approach ## (\delta_{ij} +au_{i}v_{j})(\delta_{mn} - \frac{au_{m}v_{n}}{1-au_{k}v_{k}}) ## and do the calculations? Does this term make sense ## \delta_{ij}\delta_{mn}##?

Thanks lot
 

Answers and Replies

  • #2
BvU
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Hello ##\Theta##, :welcome:

You don't want four indices, but three: one to sum over and the other two are the indices of the product matrix.

Consider matrices A, B and C: what is the expression for ##C_{ij}## in terms of ##A_{..}## and ##B_{..}## ?

PS do you mean ##A^{-1}_{ij} ## as in ##(A^{-1})_{ij} ## ?
 
  • #3
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Hello ##\Theta##, :welcome:

You don't want four indices, but three: one to sum over and the other two are the indices of the product matrix.

Consider matrices A, B and C: what is the expression for ##C_{ij}## in terms of ##A_{..}## and ##B_{..}## ?

PS do you mean ##A^{-1}_{ij} ## as in ##(A^{-1})_{ij} ## ?
I believe it is ## C_{ij}=A_{ik}B_{kj}## ? Yes you are correct it is ## A_{ij}^{-1}## my mistake. So if I keep three free indices I have something like : ## C_{ij}=(\delta_{im} +au_{i}v_{m})(\delta_{mj} - \frac{au_{m}v_{j}}{1-au_{k}v_{k}})= \delta_{im}\delta_{mj}- \delta_{im}\frac{au_{m}v_{j}}{1-au_{k}v_{k}} +\delta_{mj}au_{i}v_{m} -au_{i}v_{m}\frac{au_{m}v_{j}}{1-au_{k}v_{k}} ## ?
 
  • #4
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Einstein notation.... Sorry mate it's all about the lathes, spanners and Sir Issac around here.
 
  • #5
PeroK
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I believe it is ## C_{ij}=A_{ik}B_{kj}## ? Yes you are correct it is ## A_{ij}^{-1}## my mistake. So if I keep three free indices I have something like : ## C_{ij}=(\delta_{im} +au_{i}v_{m})(\delta_{mj} - \frac{au_{m}v_{j}}{1-au_{k}v_{k}})= \delta_{im}\delta_{mj}- \delta_{im}\frac{au_{m}v_{j}}{1-au_{k}v_{k}} +\delta_{mj}au_{i}v_{m} -au_{i}v_{m}\frac{au_{m}v_{j}}{1-au_{k}v_{k}} ## ?
Yes, that's the idea.
 
  • #6
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Yes, that's the idea.
I think I completed it. thank you.
 
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  • #7
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Einstein notation.... Sorry mate it's all about the lathes, spanners and Sir Issac around here.
I think fluids is a specialty of Mech,Eng.? and you need tensors.
 

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