Index notation/ Tensors, basic algebra questions.

[binbagsss]

Ok I have T$_{ij}$=μS$_{ij}$ + λ δ$_{ij}$δS$_{kk}$.
I am working in R^3.

(I am after S in terms of T) . I multiply by δ$_{ij}$ to attain:

δ$_{ij}$T$_{ij}$=δ$_{ij}$μS$_{ij}$ + δ$_{ij}$ λ δ$_{ij}$δT$_{kk}$

=> T$_{jj}$=δ$_{jj}$λS$_{kk}$+μS$_{jj}$ *

My question is , for the LH term of * I choose T$_{jj}$ rathen than T$_{ii}$. I then get the same decision for μS$_{jj}$ or μS $_{ii}$ on the last term on the RHS. Does this decision need to be consistent with each other?

Next/Main question...
The solution then continues to attain
T$_{kk}$=(μ+3λ)S$_{jj}$

Which I can not see how we have got to. δ$_{jj}$=3, so for RHS of * I get : 3λS$_{kk}$+μS$_{jj}$ .

I then rename j and k, to get T$_{kk}$ = 3λS$_{jj}$+μS$_{kk}$.
My 'S's' do not have the same dummy indice?

Many Thanks to anyone who can shed some light on this !

 

[qbert]

looks like you're having problems with the implicit sums. when in
doubt write it out.

$$T_{ij}=\mu S_{ij} + \lambda \delta_{ij} \sum_k \delta S_{kk}$$

You then hit by a $\delta_{ij}$ and sum over both i and j

$$\sum_{ij} \delta_{ij} T_{ij}=\mu \sum_{ij} \delta_{ij} S_{ij} + \lambda \sum_{ij} \delta_{ij} \delta_{ij} \sum_k \delta S_{kk},$$

Using the property that $\delta_{ij} = 0$ unless i = j, and is 1 if i=j, we have

$$\sum_{i} T_{ii}=\mu \sum_{i} S_{ii} + \lambda \sum_{i} \delta_{ii} \sum_k \delta S_{kk},$$

Answer 1. it doesn't matter what your dummy index is i, j, k. it's a dummy index. however if you
want to be kind to somebody reading you should make the indices match on both sides of an equation.

Answer 2. the factor of three comes from
$$\sum_{i} \delta_{ii} = \delta_{11} + \delta_{22} + \delta_{33}= 1+1+1 = 3$$

This leaves
$$\sum_{i} T_{ii}=\mu \sum_{i} S_{ii} + 3 \lambda \sum_k \delta S_{kk},$$

Does it matter that the summation indices are different? not really but it is ugly. So use
a different letter!
$$\sum_{i} T_{ii}=\mu \sum_{i} S_{ii} + 3 \lambda \sum_i \delta S_{ii},$$
$$\sum_{i} T_{ii}= \sum_{i} (\mu + 3 \lambda) S_{ii},$$

revert to the summation convention
$$T_{ii}= (\mu + 3 \lambda) S_{ii}.$$

 

[binbagsss]

Thanks a lot I see now - I thought the rules were that if i rename Skk to Sii, I must also change Tii to Tkk.