Understanding Index Notation: Allowed Combinations Explained

[squire636]

I'm not sure if this is the correct place to ask this question, so please let me know if there is a better place for me to post it. I'm having trouble understanding index notation. I understand the basics, such as in the following examples:

(a x b) = ε_ijka_jb_k

ε_ijkε_iab = δ_jaδ_kbδ_jbδ_ka

δ_ija_j = a_i

Homework Statement

Here's the problem I'm trying to solve: Which of the following are allowed in index notation:

a = b_ic_ijd_j
a = b_ic_i + d_j
a_i = δ_ijb_i + c_i
a_k = b_ic_ki

There are a whole bunch more of these, but I think I can probably figure those out if I get some help with these. Please explain which of those are allowed in index notation and why they are allowed or not allowed. Thanks!

Homework Equations

The Attempt at a Solution

I'd say that the first is allowed because neither i nor j is repeated more than twice.
Same with the second.
For the third, I know δ_ijb_i = b_j but then I'm not sure what to make of it from there.
The fourth seems okay since k is the free variable and we're summing over i.


Thanks again for your explanations!

 

[vela]

These should be:

(a x b)_i = ε_ijka_jb_k

ε_ijkε_iab = δ_jaδ_kb-δ_jbδ_ka

You have to have the same free indexes on both sides of the equation. How does that change your answers?

 

[squire636]

Ah yes, the minus sign that I forgot was simply a typo, and I had seen the cross product written with that subscript...but I didn't realize it was necessary.


Does this imply that the first two are not allowed because the a does not have a subscript i?

The third one wouldn't be allowed either because the free index on the left side is "i", but the right side will have the indices "j" and "i."

The fourth looks like it should work.



New question:
What about something like a_ij = b_ji ? I can't tell if this is fine because we're just switching the indices, or if it is complete nonsense since there is no repeated index and therefore no summing.

How about a_kl = b_ic_kid_l + e_ki ? Here, everything looks fine except for the addition of e. k and l are the free indices, and we are summing over i...but I'm not sure how to interpret the addition of e_ki.

Thanks again!

 

[vela]

Remember only the free indices need to appear on both sides. The ones that are summed over don't. In the first one, both i and j are summed over, so they don't need to appear on the lefthand side.

The fourth one is fine, as you said. The second and third aren't good, but not for the reasons you said. Based on what I said, can you see why they're wrong now?

 

[squire636]

For the second one...the first term is being summed over "i", and "j" is the free index, but it does not appear on the left side.

For the third...the first term is being summed over "i", but then the same index occurs in the second term and is not being summed, so this makes no sense.

 

[vela]

Right.

 

[squire636]

Thanks! Any comments on the new question that I posted? (see "New Question" in bold a few posts up).

 

[vela]

What do you think?

 

[squire636]

I don't actually think I can get any further in my reasoning than I already have...

 

[vela]

A summation isn't required, and you should already be able to answer the second one. I'm not sure why you're confused about it.