When writing the expression of $(\textbf a\times \textbf b)\bullet(\textbf c\times \textbf d) $ in index notation, I am told that the correct index notation is as follows:$$\epsilon_{ijk}a_jb_k\epsilon_{ilm}c_ld_m$$ Why do we relabel the indices $j,k$ but not the index $i$? Is there a general rule for this?
When to relabel dummy variables in tensor index notation
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$\begingroup$
tensor-products
cross-product
index-notation
1 Answers
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You could relabel the index $i$ too, they're all mute. What you can't do is use the index $j$ and $k$ again for the second vector, because you're already using them in the first two summations. For one second, let's write summations again.
$$\begin{align*} (a \times b)\cdot (c \times d) &= \sum_i (a \times b)_\color{blue}{i} (c \times d)_\color{blue}{i} \\ &= \sum_i \left(\sum_{j,k}\epsilon_{\color{blue}{i}jk}a_jb_k\right)\left(\sum_{\color{red}{j,k}}\epsilon_{\color{blue}{i}\color{red}{jk}}c_\color{red}{j}d_\color{red}{k}\right) \\ &= \sum_{i,j,k,l,m}\epsilon_{ijk}a_jb_k\epsilon_{ilm}c_ld_m.\end{align*}$$We still can write the step in red, but you need to relabel in the next step because $$\sum_{i,j,k,l,m}\epsilon_{ijk}a_jb_k\epsilon_{ilm}c_ld_m \neq \sum_{i,j,k}\epsilon_{ijk}a_jb_k\epsilon_{ijk}c_jd_k.$$As for relabeling $i$: $$\sum_i (a \times b)_i (c \times d)_i = \sum_p (a \times b)_p (c \times d)_p = \sum_q (a \times b)_q (c \times d)_q = \cdots$$
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0OK thank you! This clarifies other confusions I had. But the question I was aiming to get across is: why don't we label $i$ as $s$ for example. Why don't we change the index of $i$. @Ivo Terek – 2017-02-25
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0But you could. We have $$\epsilon_{ijk}a_jb_k\epsilon_{ilm}c_ld_m = \epsilon_{sjk}a_jb_k\epsilon_{slm}c_ld_m.$$You **can't** write $$\epsilon_{ijk}a_jb_k\epsilon_{slm}c_ld_m $$though, since in the first sum we are dealing with the same components of each cross product. I indicated that in blue in an edit, see if you understand. – 2017-02-25
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1Ah excellent! Thank you! – 2017-02-25
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0Glad to help! :) – 2017-02-25