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

In summary, Continuum mechanics can be confusing, so be sure to ask for help if you don't understand an equation.
  • #1
4
1
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
 
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  • #2
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
BvU said:
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
Einstein notation... Sorry mate it's all about the lathes, spanners and Sir Issac around here.
 
  • #5
Theta_84 said:
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
PeroK said:
Yes, that's the idea.
I think I completed it. thank you.
 
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Likes BvU
  • #7
xxChrisxx said:
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.
 

1. What is Einstein index notation?

Einstein index notation, also known as Einstein summation convention, is a mathematical notation used to simplify and condense expressions involving summation of vectors and tensors. It was introduced by Albert Einstein in his theory of general relativity.

2. How is Einstein index notation written?

In Einstein index notation, repeated indices are implicitly summed over, eliminating the need for explicit summation signs. The notation uses Greek letters, such as α, β, γ, to denote indices and a superscript and subscript to indicate the upper and lower index, respectively. For example, the expression Aαβ Bβ would be written as AαBα in Einstein notation.

3. What are the benefits of using Einstein index notation?

Einstein index notation simplifies and condenses mathematical expressions, making them easier to read and understand. It also allows for easier manipulation and calculation of vectors and tensors in physics and engineering problems.

4. How is Einstein index notation related to tensor calculus?

Einstein index notation is an important tool in tensor calculus, as it allows for the manipulation of tensors with a concise and elegant notation. It is particularly useful in the study of general relativity, where tensors are used to describe the curvature of spacetime.

5. Are there any limitations to using Einstein index notation?

While Einstein index notation is a powerful tool, it can become cumbersome when dealing with large numbers of indices. In addition, it may not be suitable for all mathematical expressions, and some may prefer to use traditional summation notation for clarity.

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