Tensor identity - Help

jvicens

I'm having some trouble to prove the following tensor identity shown below in Einstein's summation convention:

[tex](a_{ij}+a_{ji})x_{i}x_{j}=2a_{ij}x_{i}x_{j}[/tex]

I expanded the terms but when I did group them I didn't get the identity. The only way I could get the identity is if

[tex]a_{ij}=a_{ji}[/tex]

and I don't see a reason why this would be so.

Obviously I'm missing something. Can somebody tell what is it that I'm doing wrong?

Lonewolf

It's true that [tex]a_{ij}x_{i}x_{j} = a_{ji}x_{i}x_{j}[/tex]. Putting in the summation signs may help you to see this.

robphy

To elaborate on Lonewolf's reply...
[tex]
\begin{align*}
a_{ij}x_{i}x_{j}
&= a_{ji}x_{j}x_{i}&&\text{relabel repeated [dummy] indices}\\
&= a_{ji}x_{i}x_{j}&&\text{reorder the writing of the x factors}\\
\end{align*}
[/tex]

You'll learn that [itex]\frac{1}{2}(a_{ij} + a_{ji})[/itex]
is called "the symmetric part of [itex] a_{ij} [/itex]", and is written as [itex] a_{(ij)} [/itex].
Hence, your identity can be written
[tex] 2a_{(ij)}x_{i}x_{j} =2a_{ij}x_{i}x_{j}[/tex].

Here is an "index gymnastics" proof, starting with half of your right-hand-side:
[tex]
\begin{align*}
a_{ij}x_{i}x_{j}
&= a_{ij}x_{(i}x_{j)}&&\text{since } x_{i}x_{j} = x_{(i}x_{j)} \\
&= a_{(ij)}x_{i}x_{j}&&\text{since i and j are being symmetrized}
\end{align*}
[/tex]

jvicens

Guys, thanks to you help I got it right this time after some thinking and some calculations. I realized one of my problems was the fact that when I was looking at
[tex]a_{ij}=a_{ji}[/tex]
I was thinking of it as when we say
[tex]a=b[/tex]
The way I should look at that expression is not isolated from the rest of the other terms. The important thing, I think, is not the term itself isolated but how the summation terms comes up after the entire summation is expanded.
I can continue now with the next page of my book