Can Einstein Index Notation Help Me Solve Equations in Continuum Mechanics?

[Theta_84]

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

 

[BvU]

Hello $\Theta$,

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 $(A^{-1})_{ij}$ ?

 

[Theta_84]

Hello $\Theta$,

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 : $C_{ij}=(\delta_{im} +au_{i}v_{m})(\delta_{mj} - \frac{au_{m}v_{j}}{1-au_{k}v_{k}})= \delta_{im}\delta_{mj}- \delta_{im}\frac{au_{m}v_{j}}{1-au_{k}v_{k}} +\delta_{mj}au_{i}v_{m} -au_{i}v_{m}\frac{au_{m}v_{j}}{1-au_{k}v_{k}}$ ?

 

[xxChrisxx]

Einstein notation... Sorry mate it's all about the lathes, spanners and Sir Issac around here.

 

[PeroK]

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.

 

[Theta_84]

Yes, that's the idea.

I think I completed it. thank you.

 

[Theta_84]

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.