The Levi-Civita Symbol and its Applications in Vector Operations

[rudy]

TL;DR

Trying to check that: $e_i \times e_j = ε_{ijk}e_k$

I appear to have an inconsistency (or an error)

Hello all,

I was just introduced the Levi-Civita symbol and its utility in vector operations. The textbook I am following claims that, for basis vectors $e_1, e_2, e_3$ in an orthonormal coordinate system, the symbol can be used to represent the cross product as follows:

$$e_i \times e_j = ε_{ijk}e_k$$

So, as a check (and because I have too much time on my hands) I plugged in numbers for $i, j, k$ starting with [1 1 1], [1 1 2] ...

[1 1 1]: $e_1 \times e_1 = ε_{111}e_1$; correct because $ε_{111} = 0$
[1 1 2]: $e_1 \times e_1 = ε_{112}e_2$; correct because $ε_{112} = 0$
[1 1 3]: $e_1 \times e_1 = ε_{113}e_3$; correct because $ε_{113} = 0$

however:

[1 2 1]: $e_1 \times e_2 = ε_{121}e_1$; incorrect because $ε_{121} = 0$ and $e_1 \times e_2 = e_3$

Does anyone know what I am doing wrong? I suppose everyone knows that the cross product must be orthogonal to both vectors, and that this will void any combination of $i, j, k$ where k is equal to either i or j. However that would mean that this relationship is not valid for all cases of $i, j, k$. In other words there would need to be a subscript saying that "i and j can repeat but k must be unique" or something like that. If anyone can help explain this I'd appreciate it.

-DR

 

[PeroK]

Summary:: Trying to check that: $e_i \times e_j = ε_{ijk}e_k$

This assumes the "summation convention" where sums are suppressed. It means:
$$e_i \times e_j = \sum_{k = 1}^{3} ε_{ijk}e_k$$

 

[rudy]

Brilliant! Thanks for clearing that up

 

[PeroK]

Brilliant! Thanks for clearing that up

PS it's the double $k$ index that tells you there's a sum involved.

 

[rudy]

I see, thanks. So my terms in brackets should really read [1 1 k]; [1 2 k]; etc

 

[PeroK]

I see, thanks. So my terms in brackets should really read [1 1 k]; [1 2 k]; etc

It's actually only $9$ equations, not $27$. What it really, really means is:
$$e_i \times e_j = \sum_{k = 1}^{3} ε_{ijk}e_k \ \ (i, j = 1, 2, 3)$$

 

[rudy]

Appreciate your patience! Still practicing this summation convention notation.

 

[etotheipi]

Appreciate your patience! Still practicing this summation convention notation.

I feel your pain, I've been trying to grok this recently also. This page has some good examples.

 

[rudy]

Glad I'm not the only one struggling lol. Thanks for the reference.

 

[etotheipi]

A lot of the questions I've been doing only require a few different tricks. The most common ones I've come across are $$[\vec{u} \times \vec{v}]_i = \epsilon_{ijk} u_j v_k$$ $$\vec u \times \vec v = \epsilon_{ijk} u_j v_k \vec e_i$$ $$\nabla \times \vec v = \epsilon_{ijk} \vec e_i \nabla_j v_k$$ and then some Delta function ones like $$\epsilon_{ijk}\epsilon_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}$$ $$\epsilon_{ijk}\epsilon_{ljk} = 2\delta_{il}$$ $$\epsilon_{ijk}\epsilon_{ijk} = 6$$And finally some that that are much more easily remembered, like $\vec u \cdot \vec v = u_i v_i$.