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# Einstein Summation convention with vectors watch

1. Just trying to become fluent with using these in questions, with the first one that I'm stuck on being:

Using Einstein Summation convention, prove that

I began with simplifying a bit and saying that so that we then have:

and now I THINK (because I'm not certain) I can express this as and now I'm not sure if there is anything here to be simplified or not, or where to move to - do I need to write out ALL these (non-zero) terms? Perhaps the epsilons go to some Kronecker delta? Though I don't see how that might be possible.

ALSO: I believe here with
2. (Original post by RDKGames)
..
Couple of things:

I assume r is supposed to equal i.e. it is the spatial position.

In which case and then .

The other thing I would do (*) is replace by ;

(*) if you want to do this using summation convention you will basically need to use the identity (which should have all been covered in lectures).
3. (Original post by RDKGames)
Perhaps the epsilons go to some Kronecker delta? Though I don't see how that might be possible.

[edit: nvm, DFrank got there first]
4. (Original post by Zacken)

[edit: nvm, DFrank got there first]
Thanks for providing a version of the identity that matches RDK's indices...
5. (Original post by RDKGames)
...do I need to write out ALL these (non-zero) terms....
Just for info: with practice you can often do these directly from the vector notation, or if you do need to go to summation convention, it's very rare to need to write out individual terms.

For instance, I can tell by looking that , from which the result follows.
6. It is probably a good exercise to prove , which is essentially what you do, in the special case u = c, v=r , w = u = c by using the identity in DFrank's post.
7. (Original post by DFranklin)
Couple of things:

I assume r is supposed to equal i.e. it is the spatial position.

In which case and then .
Ah yes, that is mentioned at the top of the worksheet that I missed out. So

The other thing I would do (*) is replace by ;
Cheers, I'll have a go using this.

(*) if you want to do this using summation convention you will basically need to use the identity (which should have all been covered in lectures).
(Original post by Zacken)

[edit: nvm, DFrank got there first]
Thanks. Yes this formula was derived in the lecture notes, I just got stuck on applying it correctly as the first index did not start with on both, then I quickly realised that which made the application to Kronecker delta simple.

(Original post by DFranklin)
Just for info: with practice you can often do these directly from the vector notation, or if you do need to go to summation convention, it's very rare to need to write out individual terms.

For instance, I can tell by looking that , from which the result follows.
I see. Currently I'm not a fan of this convention but I need to get used to it, and it took me a short while to see how you got those but they are relatively simple so I hope soon enough they'll just come out naturally for me.
8. (Original post by Zacken)
It is probably a good exercise to prove , which is essentially what you do, in the special case u = c, v=r , w = u = c by using the identity in DFrank's post.
Although one feels something has gone very wrong with the planning of the course if this hasn't been explictly covered in lectures.
9. (Original post by Zacken)
It is probably a good exercise to prove , which is essentially what you do, in the special case u = c, v=r , w = u = c by using the identity in DFrank's post.
(Original post by DFranklin)
Although one feels something has gone very wrong with the planning of the course if this hasn't been explictly covered in lectures.
We've covered this and proved it, just completely forgot about it while doing this question.

Prove that

So I seem to have lost one of them along the way but I can't pinpoint where. I think it's the part at the end where I say (not explicitly) that but isn't this true?
11. (Original post by RDKGames)
So I seem to have lost one of them along the way but I can't pinpoint where. I think it's the part at the end where I say (not explicitly) that but isn't this true?
No; implicitly i and j range from 1 to 3; and when i = j.

More explictly, (*) (as I posted upstream).

I know you're just learning, but this is an absolutely fundamental relationship - you'll be using (*) more than any other rule (other than how to use deltas at all) as you do these questions.

Edit: noticed another mistake. In the same line, it looks like you have thought that (somehow!). Again if you use (*) you see that it actually equals , which ends up being a constant.

Spoiler:
Show

12. (Original post by DFranklin)
No; implicitly i and j range from 1 to 3; and when i = j.

More explictly, (*) (as I posted upstream).

I know you're just learning, but this is an absolutely fundamental relationship - you'll be using (*) more than any other rule (other than how to use deltas at all) as you do these questions.

Edit: noticed another mistake. In the same line, it looks like you have thought that (somehow!). Again if you use (*) you see that it actually equals , which ends up being a constant.

Spoiler:
Show

Ah, got it - I was slightly misunderstanding the delta thing there.

Thanks.
13. (Original post by RDKGames)
Ah, got it - I was slightly misunderstanding the delta thing there.

Thanks.
Actually, I kind of mislead you on this; what you've done still isn't right. (Note that I'm using x_i instead of r_i everywhere; to be honest I think using r_i is "essentially" wrong - that is, it's only right because r_i = x_i, and you're implicitly using this throughout, so you should just write x_i (or differentiate w.r.t. r_i, equivalently)).

It's true that , but it is not true that

What you need to do is use the product rule (for differentiation):

You need to do similarly with the expression - it's a product, and the partial derivative acts on both terms ( and ).
14. (Original post by DFranklin)
Actually, I kind of mislead you on this; what you've done still isn't right. (Note that I'm using x_i instead of r_i everywhere; to be honest I think using r_i is "essentially" wrong - that is, it's only right because r_i = x_i, and you're implicitly using this throughout, so you should just write x_i (or differentiate w.r.t. r_i, equivalently)).

It's true that , but it is not true that

What you need to do is use the product rule (for differentiation):

You need to do similarly with the expression - it's a product, and the partial derivative acts on both terms ( and ).
Oh okay, thanks for clarifying. It makes sense what you've said, and I've adjusted my answer. The caught me out as it's hard for me to keep track of the fact that these are sums, so I was stuck here confused as I kept getting =0 from me writing

15. DFranklin

With and being scalar and vector fields respectively, prove that using E.C.

So I began by saying that:

which by the product rule gives

I can see how the second term comes to give but I can't quite see how the first term goes to
16. (Original post by RDKGames)
DFranklin

With and being scalar and vector fields respectively, prove that using E.C.

So I began by saying that:

which by the product rule gives

I can see how the second term comes to give but I can't quite see how the first term goes to
I think the question you need to ask yourself is what does actually mean? (At which point you'll realise you don't have much more to do...)
17. (Original post by DFranklin)
I think the question you need to ask yourself is what does actually mean? (At which point you'll realise you don't have much more to do...)
Oh right, so it's just as simple as ?
18. Yes, basically.

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