Einstein indicial notation problem

hello

I am new to the forum so I don't know if I am posting in the right section :)

I want to prove the following equality in index notation :

S[i,k]S[k,j]S[i,l]S[l,j]=1/2*S[i,j]S[i,j]S[k,l]S[k,l] .

S is a symmetric second-order tensor with property S[i,i] = 0 .

I have been struggling with this for a week ,any help is appreciated, thanks in advance :)

Re: Einstein indicial notation problem

Hey milad.

If something is symmetric then it means that S[i,k] = S[k,i]. Did you try and expand the summation in an explicit way and not in the shorthand Einstein notation?

Re: Einstein indicial notation problem

I expanded the equation using matlab and I got :

S[i,k]S[k,j]S[i,l]S[l,j]-1/2*S[i,j]S[i,j]S[k,l]S[k,l]= s[i,i](~)

which means if we apply the property s[i,i]= 0 and symmetry they are equal .

I also checked this numerically for over 10 different set numbers and if you have symmetry and S[i,i]=0 ,then they are equal .

Re: Einstein indicial notation problem

Is that all you need to do or do you need to prove it symbolically?

Re: Einstein indicial notation problem

I need to prove it symbolically :)

Re: Einstein indicial notation problem

The first thing I would do is to convert from Einstein notation to actually sigma sums.

Re: Einstein indicial notation problem

tried that ,I couldn't solve it :( .

Re: Einstein indicial notation problem

What do you mean by $\displaystyle S_{[i,j]}$? The component $\displaystyle (i,j)$ of $\displaystyle S$ or the antisymmetrization $\displaystyle S_{[i,j]}=\frac{1}{2}(S_{ij}-S_{ji})$

Re: Einstein indicial notation problem

I mean (i,j) component . antisymmetrization of a symmetric tensor will be zero .

Re: Einstein indicial notation problem

By S[i,i] = 0 do you mean diagonal elements are zero or sum of diagonal elements is zero?

Re: Einstein indicial notation problem

sum of diagonal elements are zero

Re: Einstein indicial notation problem

What do you mean by different set of numbers? There are no free indices in your expression, there is one set of numbers only, say the range of values that $\displaystyle i,j,k,l$ can assume.

Re: Einstein indicial notation problem

I meant I checked the expression numerically ,I assigned different sets of numbers to elements of the tensor S , and checked the answers. two sides of the equation are equal if the values assigned to S have those two properties(symmetry , S[i,i]=0 ) . :)

Re: Einstein indicial notation problem

I solved this and wrote it up in Word because of all the indices. I did a copy paste in Latex help and everything copied correctly. When I tried to put it in this thread, none of the subscripts and superscripts copied over and to do it all manually is a nightmare. So I copied the solution to Latex help and give the link here.

http://mathhelpforum.com/latex-help/...tml#post782895

I would appreciate it if someone put the solution in this post directly. OK, did copy paste from Latex post and that worked.

OP: I want to prove the following equality in index notation :

S[i,k]S[k,j]S[i,l]S[l,j]=1/2*S[i,j]S[i,j]S[k,l]S[k,l] .

let S =|s_{ij}| and S_{ij} ≡ s_{ij}

Prove:

S^{4}_{ii} = ½ (S^{2}_{jj})(S^{2}_{kk}) *

S is symmetric so there is a coordinate system in which S is diagonal with real components. In this coordinate system:

S=diag(s_{11},s_{22},s_{33})

S^{2}=diag(s_{11}^{2},s_{22}^{2},s_{33}^{2})

S^{4}=diag(s_{11}^{4},s_{22}^{4},s_{33}^{4})

2dim proof (3dim is the same with more algebra. You can’t get beyond this point with summation convention):

s_{11} + s_{22} = 0, sum of diagonal elements is a tensor invariant

s_{11}^{2} + s_{22}^{2} = -2s_{11}s_{22}

(s_{11}^{2} + s_{22}^{2}) (s_{11}^{2} + s_{22}^{2}) = 4s_{11}^{2}s_{22}^{2}

s_{11}^{4} + s_{22}^{4} = 2s_{11}^{2}s_{12}^{2} = ½ (s_{11}^{2} + s_{22}^{2}) (s_{11}^{2} + s_{22}^{2})

S^{4}_{ii} = ½ (S_{jj})(S_{kk})

Proof holds in any coordinate system because contraction of a tensor is a tensor.

* s_{ik}s_{kj}s_{il}s_{lj} = S^{2}_{ij}S^{2}_{ij} = S^{4}_{ii}

(A_{ij}A_{jk} = A^{2}_{ik}, A_{ij}A_{ji} = A^{2}_{ii})

Re: Einstein indicial notation problem

thanks .I will try to do the same in 3D .