(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

How to proof the following property of tensor invariants?

Where:

##[\mathbf{a\; b\; c}]=\mathbf{a\cdot (b\times c)} ##,

##\mathbf{T} ##is a second order tensor,

##\mathfrak{J}_{1}^{T}##is its first invariant,

##\mathbf{u, v, w}## are vectors.

2. Relevant equations

$$[\mathbf{T\cdot u\; v\; w}]+[\mathbf{u\; T\cdot v\; w}]+[\mathbf{u\; v\; T\cdot w}]=\mathfrak{J}_{1}^{T}[\mathbf{u\; v\; w}]$$

3. The attempt at a solution

$$T^{l}{ }_{i}u^{i}v^{j}w^{k}\epsilon_{ljk}+T^{l}{ }_{j}u^{i}v^{j}w^{k}\epsilon_{ilk}+T^{l}{ }_{k}u^{i}v^{j}w^{k}\epsilon_{ijl}

$$$$=1/6(T^{l}{ }_{i}u^{i}v^{j}w^{k}\epsilon_{ljk}\epsilon_{\alpha \beta \gamma }\epsilon^{\alpha \beta \gamma }+T^{l}{ }_{i}u^{i}v^{j}w^{k}\epsilon_{ilk}\epsilon_{\alpha \beta \gamma }\epsilon^{\alpha \beta \gamma }+T^{l}{ }_{i}u^{i}v^{j}w^{k}\epsilon_{ijl}\epsilon_{\alpha \beta \gamma }\epsilon^{\alpha \beta \gamma })=???$$

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# Homework Help: A proof about tensor invariants

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