- #1
binbagsss
- 1,326
- 12
Ok I have T[itex]_{ij}[/itex]=μS[itex]_{ij}[/itex] + λ δ[itex]_{ij}[/itex]δS[itex]_{kk}[/itex].
I am working in R^3.
(I am after S in terms of T) . I multiply by δ[itex]_{ij}[/itex] to attain:
δ[itex]_{ij}[/itex]T[itex]_{ij}[/itex]=δ[itex]_{ij}[/itex]μS[itex]_{ij}[/itex] + δ[itex]_{ij}[/itex] λ δ[itex]_{ij}[/itex]δT[itex]_{kk}[/itex]
=> T[itex]_{jj}[/itex]=δ[itex]_{jj}[/itex]λS[itex]_{kk}[/itex]+μS[itex]_{jj}[/itex] *
My question is , for the LH term of * I choose T[itex]_{jj}[/itex] rathen than T[itex]_{ii}[/itex]. I then get the same decision for μS[itex]_{jj}[/itex] or μS [itex]_{ii}[/itex] 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[itex]_{kk}[/itex]=(μ+3λ)S[itex]_{jj}[/itex]
Which I can not see how we have got to. δ[itex]_{jj}[/itex]=3, so for RHS of * I get : 3λS[itex]_{kk}[/itex]+μS[itex]_{jj}[/itex] .
I then rename j and k, to get T[itex]_{kk}[/itex] = 3λS[itex]_{jj}[/itex]+μS[itex]_{kk}[/itex].
My 'S's' do not have the same dummy indice?
Many Thanks to anyone who can shed some light on this !
I am working in R^3.
(I am after S in terms of T) . I multiply by δ[itex]_{ij}[/itex] to attain:
δ[itex]_{ij}[/itex]T[itex]_{ij}[/itex]=δ[itex]_{ij}[/itex]μS[itex]_{ij}[/itex] + δ[itex]_{ij}[/itex] λ δ[itex]_{ij}[/itex]δT[itex]_{kk}[/itex]
=> T[itex]_{jj}[/itex]=δ[itex]_{jj}[/itex]λS[itex]_{kk}[/itex]+μS[itex]_{jj}[/itex] *
My question is , for the LH term of * I choose T[itex]_{jj}[/itex] rathen than T[itex]_{ii}[/itex]. I then get the same decision for μS[itex]_{jj}[/itex] or μS [itex]_{ii}[/itex] 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[itex]_{kk}[/itex]=(μ+3λ)S[itex]_{jj}[/itex]
Which I can not see how we have got to. δ[itex]_{jj}[/itex]=3, so for RHS of * I get : 3λS[itex]_{kk}[/itex]+μS[itex]_{jj}[/itex] .
I then rename j and k, to get T[itex]_{kk}[/itex] = 3λS[itex]_{jj}[/itex]+μS[itex]_{kk}[/itex].
My 'S's' do not have the same dummy indice?
Many Thanks to anyone who can shed some light on this !