# Index notation identity

1. Nov 10, 2015

### Incand

1. The problem statement, all variables and given/known data
The antisymmetric tensor is constructed from a vector $\vec a$ according to $A_{ij} = k\varepsilon_{ijk}a_k$.
For which values of $k$ is $A_{ij}A_{ij} = |\vec a|^2$?

2. Relevant equations
Identity
$\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$

3. The attempt at a solution
$A_{ij}A_{ij} = k^2\varepsilon_{ijk}\varepsilon_{ijm}a_ka_m = k^2\varepsilon_{jki}\varepsilon_{ijm}a_ka_m = k^2(\delta_{jj}\delta_{km}-\delta_{jm}\delta_{kj})a_ka_m = k^2(\delta_{km}-\delta_{km})a_ka_m =0$
Which I obviously shouldn't get but I can't see where I'm making an error.

2. Nov 10, 2015

### fzero

You are using the summation convention, so
$$\delta_{jj} \equiv \sum_{j=1}^3 \delta_{jj}.$$
This gives a different numerical factor in front of the first $\delta_{km}$ term.

3. Nov 10, 2015

### Incand

Right thanks!