Prove a tensor identity

Question

Prove the tensor identity: $$e_{ij}^2-\frac{1}{3}e_{kk}^{2}=(e_{ij}-\frac{1}{3}\delta_{ij}e_{kk})^{2}$$

This equality we used it to prove the conservation of energy equation although i don't see clearly how to prove it

Answer 1

As per @Deane ’s comment: expanding R.H.S.

\begin{align*} (e_{ij} - \frac{1}{3}\delta_{ij}e_{kk})^2 &= (e_{ij} - \frac{1}{3}\delta_{ij}e_{kk}) (e_{ij} - \frac{1}{3}\delta_{ij}e_{kk}) \\ &= e_{ij} e_{ij} - \frac{2}{3}e_{ij}\delta_{ij}e_{kk} + \frac{1}{9} \delta_{ij} \delta_{ij}e_{kk} e_{kk} \\ &= e_{ij}^2 - \frac{2}{3}e_{jj}e_{kk} + \frac{1}{9} \delta_{jj} e_{kk}^2 \\ &= e_{ij}^2 - \frac{2}{3}e_{kk}e_{kk} + \frac{1}{9} (3) e_{kk}^2 & (\delta_{jj} = n, n = \text{dimensions})\\ &= e_{ij}^2 - \frac{2}{3}e_{kk}^2 + \frac{1}{3} e_{kk}^2 = e_{ij}^2 - \frac{1}{3}e_{kk}^2\\ \end{align*}

Answer 2

The summation convention works most nicely when you are manipulating multilinear expresions, so you can translate everyting to multilinear terms and work from there. For example, $e_{ij}^2$ is notation for $e_{ij}\delta^{ik}\delta^{jl}e_{kl}$, while $e_{kk}$ really means $e_{kl}\delta^{kl}$ and $e_{kk}^2$ means $e_{kl}\delta^{kl}e_{mn}\delta^{mn}$. With this in mind, write the right hand side $(e_{ij}-\frac{1}{3}\delta_{ij}e_{kk})^{2}$ as $$ (e_{ij}-\frac{1}{3}\delta_{ij}e_{kr}\delta^{kr})\delta^{im}\delta^{jn} (e_{mn}-\frac{1}{3}\delta_{mn}e_{tu}\delta^{tu}) $$ and see if you can continue from there. Your goal will be to get to $$ e_{ij}\delta^{ik}\delta^{jl}e_{kl} - \frac 1 3 e_{kl}\delta^{kl}e_{mn}\delta^{mn}. $$