Einstein notation and the permutation symbol

[rcummings89]

Homework Statement

This is my first exposure to Einstein notation and I'm not sure if I'm understanding it entirely. Also I added this class after my instructor had already lectured about the topic and largely had to teach myself, so I ask for your patience in advance...

The question is:
Evaluate the following expression: ε_ijka_ia_j

Homework Equations

a ^ b = a_i e_i ^ b_j e_j = a_ib_j (e_i ^ e_i) = a_ib_j ε_ijk e_k

Where I'm following his notation that ^ represents the cross product of the two vectors

The Attempt at a Solution

Now, just going off what I have seen so far in the handout he has posted, I believe the answer to be

ε_ijka_ia_j = (a ^ a)_k or, ε_ijka_ia_j is the k^th component of a ^ b and because the expression is a vector crossed with itself it is equal to zero

But what does it mean to be the k^th component of a cross product? Honestly I'm working backward from a similar to an example he has in the handout and making the assumption that the reason the e_k component is absent from the expression is because it is the k^th component of the cross product, but from what I have to reference I cannot say with any degree of certainty if that is true and it makes me uncomfortable. Any help is greatly appreciated.

 

[pasmith]

Homework Statement

This is my first exposure to Einstein notation and I'm not sure if I'm understanding it entirely. Also I added this class after my instructor had already lectured about the topic and largely had to teach myself, so I ask for your patience in advance...

The question is:
Evaluate the following expression: ε_ijka_ia_j

Homework Equations

a ^ b = a_i e_i ^ b_j e_j = a_ib_j (e_i ^ e_i) = a_ib_j ε_ijk e_k

Where I'm following his notation that ^ represents the cross product of the two vectors

The Attempt at a Solution

Now, just going off what I have seen so far in the handout he has posted, I believe the answer to be

ε_ijka_ia_j = (a ^ a)_k or, ε_ijka_ia_j is the k^th component of a ^ b and because the expression is a vector crossed with itself it is equal to zero

But what does it mean to be the k^th component of a cross product? Honestly I'm working backward from a similar to an example he has in the handout and making the assumption that the reason the e_k component is absent from the expression is because it is the k^th component of the cross product, but from what I have to reference I cannot say with any degree of certainty if that is true and it makes me uncomfortable. Any help is greatly appreciated.

For a vector, $a_k$ is the component corresponding to $\mathbf{e}_k$. Thus $\mathbf{a} = a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 + a_3\mathbf{e}_3$. If you work out the cross product of $\mathbf{a}$ and $\mathbf{b}$ you'll find that the $\mathbf{e}_1$ component is $a_2b_3 - a_3b_2 = \epsilon_{ij1}a_i b_j$ and similarly for the other components. Thus $\epsilon_{ijk}a_i b_j$ is the $\mathbf{e}_k$ component of $\mathbf{a} \times \mathbf{b}$.

You can get that $\epsilon_{ijk}a_ia_j = 0$ more easily by observing that swapping the dummy indices $i$ and $j$ changes the sign of $\epsilon_{ijk}$ but doesn't change the sign of $a_ia_j$; thus $\epsilon_{ijk}a_ia_j = \epsilon_{jik}a_ja_i = -\epsilon_{ijk}a_ia_j = 0$. The same argument shows that if $T_{ij}$ is any symmetric tensor then $\epsilon_{ijk}T_{ij} = 0$.

 

[rcummings89]

pasmith,
Thank you for clarifying, that definitely helps! But it does bring up another question for me though; again, I'm in the early stages of learning about this notation, and know that switching the indices changes ε_ijk to ε_jik = -ε_ijk, but why does it equal zero?