# Proving vector identity by Einstein summation

Given that $$N_{ij}=\delta_{ij}-\epsilon_{ijk}n_k+n_in_j$$ and $$M_{ij}=\delta_{ij}+\epsilon_{ijk}n_k$$ Show that $$N_{ij}M_{jk}=2\delta_{ik}$$ (where $n$ is a unit vector in $R^3$)

What I have tried: $$M_{jk}=\delta_{jk}+\epsilon_{jkk}n_k=\delta_{jk}$$ So $$N_{ij}M_{jk}=(\delta_{ij}-\epsilon_{ijk}n_k+n_in_j)\delta_{jk}=\delta_{ik}-0+n_in_k$$

I'm not entirely sure what I did is legal under summation rules, but the terms I ended up doesn't match with the provided answer. Any hint is appreciated.

And what are the usual strategies when dealing with proving vector identities using suffix notation?

## 2 Answers

As far as computations go, spaceisdarkgreen's answer above is pretty ok for me. But I have some qualms with this exercise in general. Your mistake is one of the reasons why I prefer to write vector components with upper indices, and Einstein's convention is: if the same index appears twice, once up and once down, sum over it. If $n$ is a unit vector in $\Bbb R^3$, we'd write its components as $n^1$, $n^2$ and $n^3$. This calls for the index balance $$N^{ij} = \delta^{ij} - \epsilon_{\;\;k}^{ij}n^k + n^in^j \qquad \mbox{and} \qquad M^{ij} = \delta^{ij} + \epsilon_{\;\;k}^{ij}n^k.$$Then, as it is, the expression $N^{ij}M^{jk}$ makes no sense as the index $j$ appears twice up.

As it is written, $N^{ij}$ and $M^{ij}$ are components of tensors $M,N\colon (\Bbb R^3)^* \times (\Bbb R^3)^* \to \Bbb R$, in the standard basis of $\Bbb R^3$ (and its dual basis).

In order to make sense of the expression $N^{ij}M^{jk}$, we should work with the tensor $\widetilde{M}\colon \Bbb R^3 \times (\Bbb R^3)^* \to \Bbb R$, which is equivalent to $M$ under index lowering. Since (if?) these components are with respect with an orthonormal basis, and the scalar product is positive-definite, $\widetilde{M}$'s components are just $$M_i^{\;j} = \delta_i^{\;j} + \epsilon_{i\;k}^{\;j}n^k,$$and so the expression $N^{ij}M_{j}^{\;k}$ makes sense. So Einstein's convention comes with a built-in error detector. Notice that $i$ and $k$ are fixed indices. So they're off limits when we want to relabel mute indices. We then have: \begin{align}N^{ij}M_{j}^{\;k} &= (\delta^{ij} - \epsilon_{\;\;r}^{ij}n^r + n^in^j)(\delta_j^{\;k} + \epsilon_{j\;s}^{\;k}n^s) \\ &= \delta^{ij}\delta_j^{\;k} + \delta^{ij}\epsilon_{j\;s}^{\;k}n^s - \epsilon^{ij}_{\;\;r}n^r\delta_j^{\;k} - \epsilon^{ij}_{\;\;r}n^r\epsilon_{j\;s}^{\;k}n^s + n^in^j\delta_j^{\;k} + \epsilon_{j\;s}^{\;k}n^in^jn^s \\ &= \delta^{ik} + \delta^{ij}\epsilon_{j\;s}^{\;k}n^s - \epsilon^{ik}_{\;\;r}n^r - \color{blue}{(\epsilon^{ij}_{\;\;r}\epsilon_{j\;s}^{\;k})}n^rn^s + n^in^k + \color{red}{(\epsilon_{j\;s}^{\;k}n^jn^s)}n^i .\end{align}

To simplify the piece in blue, we use some well-known permutation identities: $$\epsilon^{ij}_{\;\;r}\epsilon_{j\;s}^{\;k} = -\epsilon^{ji}_{\;\;r}\epsilon_{j\;s}^{\;k} = -(\delta^{ik}\delta_{rs} - \delta^i_{\;s}\delta^k_{\;r}) = \delta^i_{\;s}\delta^k_{\;r}-\delta^{ik}\delta_{rs}.$$

Modulo sign, the piece in red is the $k$-th component of the cross product $n \times n$, which is zero. $\require{cancel}$ So: \begin{align}N^{ij}M_{j}^{\;k} &= \delta^{ik} + \delta^{ij}\epsilon_{j\;s}^{\;k}n^s - \epsilon^{ik}_{\;\;r}n^r - \delta^i_{\;s}\delta^k_{\;r}n^rn^s+\delta^{ik}\delta_{rs}n^rn^s+n^in^k \\ &=\delta^{ik} + \delta^{ij}\epsilon_{j\;s}^{\;k}n^s - \epsilon^{ik}_{\;\;r}n^r - \cancel{n^kn^i}+\delta^{ik}\color{green}{\delta_{rs}n^rn^s}+\cancel{n^in^k} \\ &=2\delta^{ik} + \delta^{ij}\epsilon_{j\;s}^{\;k}n^s - \epsilon^{ik}_{\;\;r}n^r , \end{align}since the piece in green is nothing more than $n \cdot n = 1$.

Further simplification will violate Einstein's convention as I have stated it. Since we're in a good situation (as explained earlier), you can lower all indexes and use Einstein's convention as you're using there to simplify what is left, if you want to.

For readers who know portuguese, I happen to have written some material on tensors, it might be helpful.

I just realized that we actually can simplify further, using that $r$ in the last term is mute. We'll have \begin{align}N^{ij}M_{j}^{\;k} &= 2\delta^{ik} + \delta^{ij}\epsilon_{j\;s}^{\;k}n^s - \epsilon^{ik}_{\;\;r}n^r \\ &= 2\delta^{ik} + \cancel{\epsilon_{\;\;s}^{ik}n^s} - \cancel{\epsilon^{ik}_{\;\;s}n^s} \\ &= 2\delta^{ik}.\end{align}

Your expression for $M_{jk}$ is wrong. Remember that $k$ is a dummy index in the original expression for $M_{ij}$ so if you want to relabel to make $k$ a free index, you need to replace it with something else. So you have $$M_{jk} = \delta_{jk} + \epsilon_{jkl}n_l \ne \delta_{jk}.$$

Generally speaking, simply relabeling indices shouldn't cause drastic changes to the form of an expression. $i,$ $j,$ and $k$ are just names.

More detail

It might be more instructive to eschew the efficient notation and write it out in full detail. By the summation convention, $$M_{ij} = \delta_{ij} + \sum_{k=1}^3\epsilon_{ijk}n_k = \delta_{ij} + \epsilon_{ij1}n_1 + \epsilon_{ij2}n_2 + \epsilon_{ij3}n_3.$$ So we can compute the various components of $M_{ij}$ by plugging in $1,2$ and $3$ if for $i$ and $j$ in various combinations. For instance, $$M_{11} = 1 + 0 + 0 + 0 = 1\\M_{12} = 0 + 0+ 0 +n_3 = n_3,$$ etc.

Now it should be reasonably clear by substituting $j$ for $i$ and $k$ for $j,$ that $$M_{jk} = \delta_{jk} + \epsilon_{jk1}n_1 + \epsilon_{jk2}n_2 + \epsilon_{jk3}n_3.$$ If we want to rewrite this using the efficient notation, we can just write the last three terms as a sum over an index. We should pick a letter that isn't $j$ or $k$ to avoid confusion... how about $l$ (we could pick $i$ as well for this expression, but we want to avoid this in preparation for multiplying by $N_{ij}$). So we can rewrite it as $$M_{jk} = \delta_{jk} +\sum_{l=1}^3 \epsilon_{jkl}n_l = \delta_{jk}+\epsilon_{jkl}n_l$$ where in the second equality we dropped the summation sign in accordance with the summation convention.

• Can you explain a bit more about changing indices when multiplying N and M? I'm still a bit confused. – Steve Nov 1 '17 at 1:49
• @Steve added more detail – spaceisdarkgreen Nov 1 '17 at 2:10