Tensor notation for vector product proofs

[skate_nerd]

I am new to tensor notation, but have known how to work with vector calculus for a while now. I understand for the most part how the Levi-Civita and Kronecker Delta symbol work with Einstein summation convention. However there are a few things I'm iffy about.
For example, I have a problem where I am to prove
$$\vec{A}\bullet\vec{B}\times\vec{C} = \vec{B}\bullet\vec{C}\times\vec{A} = \vec{C}\bullet\vec{A}\times\vec{B}$$
using tensor notation to avoid having to write out all the terms.
So I know the very left side of this equation would look like
$$\vec{A}\bullet(\vec{B}\times\vec{C}) = A_i (\vec{B}\times\vec{C})_i = A_i \varepsilon_{ijk} B_j C_k$$
But then I get confused when trying to assign the indices for the next two parts of the equation.
Would the second part look like this:
$$\vec{B}\bullet(\vec{C}\times\vec{A}) = B_j (\vec{C}\times\vec{A})_j = B_j \varepsilon_{jkl} C_k A_i$$
Or would the indices of the epsilon be the same as for the first part (\(\varepsilon_{ijk}\))?
Same confusion goes for the first part. The reason I have this uncertainty in my mind is because I know with the triple vector product, you have to introduce 2 extra indices. So I guess my lack of complete understanding of these functions is leaving me confused with my problem. Thanks in advance for any guidance.

 

[Dustinsfl]

I am new to tensor notation, but have known how to work with vector calculus for a while now. I understand for the most part how the Levi-Civita and Kronecker Delta symbol work with Einstein summation convention. However there are a few things I'm iffy about.
For example, I have a problem where I am to prove
$$\vec{A}\bullet\vec{B}\times\vec{C} = \vec{B}\bullet\vec{C}\times\vec{A} = \vec{C}\bullet\vec{A}\times\vec{B}$$
using tensor notation to avoid having to write out all the terms.
So I know the very left side of this equation would look like
$$\vec{A}\bullet(\vec{B}\times\vec{C}) = A_i (\vec{B}\times\vec{C})_i = A_i \varepsilon_{ijk} B_j C_k$$
But then I get confused when trying to assign the indices for the next two parts of the equation.
Would the second part look like this:
$$\vec{B}\bullet(\vec{C}\times\vec{A}) = B_j (\vec{C}\times\vec{A})_j = B_j \varepsilon_{jkl} C_k A_i$$
Or would the indices of the epsilon be the same as for the first part (\(\varepsilon_{ijk}\))?
Same confusion goes for the first part. The reason I have this uncertainty in my mind is because I know with the triple vector product, you have to introduce 2 extra indices. So I guess my lack of complete understanding of these functions is leaving me confused with my problem. Thanks in advance for any guidance.

Even if your re-arrange and your indices are in some order say jik, you can always let this new indices be lmn and then let lmn = ijk since the indices are arbitrary so you are right back to ijk.

Is that what you were asking? That is what I thought from the question. If not, clarify were I lost the point.

 

[skate_nerd]

I'm not sure I made myself clear enough...
What I am unsure of in constructing this equation in tensor notation, is if I take a cross product of two arbitrary vectors B and C, that would be in a certain arbitrary plane. Would the indices of the epsilon symbol be different than if I took the cross product of vectors C and A? Or would it make sense to call them both just \(\varepsilon_{ijk}\)? That just somehow doesn't seem like it would make sense to me, but I'm not sure what else would be correct.

 

[Dustinsfl]

I will explain the first equality which may help:
\[
\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) = \varepsilon_{ijk}a_ib_jc_k
\]
which is just a summation and a, b, c are the components of the vectors.

We can then write \(\varepsilon_{ijk}a_ib_jc_k = \varepsilon_{jki}b_jc_ka_i\).

We can either do a subsitution or recall that jki doesn't introduce a negative sign since we are still reading right to left. Thus, we can re-write the the equation as
\[
\varepsilon_{ijk}b_ic_ja_k = \mathbf{B}\cdot(\mathbf{C}\times\mathbf{A})
\]

 

[skate_nerd]

Ahhh wow thank you so much. Now I see what you meant by the indices being arbitrary. And seeing now that this proof works helps me conceptualize better what these indices are actually doing.

 

[skate_nerd]

One other quick question, about a slightly different thing if you don't mind...How would you construct a problem in vector notation and find the magnitude of that tensor product? Like for instance the vector A crossed with the vector B? Would it make sense just to dot the product with itself?

 

[Dustinsfl]

One other quick question, about a slightly different thing if you don't mind...How would you construct a problem in vector notation and find the magnitude of that tensor product? Like for instance the vector A crossed with the vector B? Would it make sense just to dot the product with itself?

I don't quite understand your question. Can you give me an example problem or question?

 

[skate_nerd]

I want to do this in tensor notation:
$$|\vec{A}\times\vec{B}|$$
Magnitude of a cross product of two arbitrary vectors. So the way I know to start is:
$$|\varepsilon_{ijk}A_j B_k|$$
To take this magnitude in vector notation is what I am not sure I understand how to do. Would it make sense to write
$$(\varepsilon_{ijk}A_j B_k)(\varepsilon_{ijk}A_j B_k)$$
and then use the "epsilon killer" identity to simplify it? Not really sure of any other way to notate it.

 

[Dustinsfl]

I want to do this in tensor notation:
$$|\vec{A}\times\vec{B}|$$
Magnitude of a cross product of two arbitrary vectors. So the way I know to start is:
$$|\varepsilon_{ijk}A_j B_k|$$
To take this magnitude in vector notation is what I am not sure I understand how to do. Would it make sense to write
$$(\varepsilon_{ijk}A_j B_k)(\varepsilon_{ijk}A_j B_k)$$
and then use the "epsilon killer" identity to simplify it? Not really sure of any other way to notate it.

I would just interpret as
\[
\lvert\mathbf{A}\times\mathbf{B}\rvert = \lVert \mathbf{A}\rVert\lVert \mathbf{B}\rVert\sin(\theta)
\]
Is there a specific end goal of this question? Should you come up with a certain expression?

 

[skate_nerd]

Well actually the goal of the question is to prove that expression you just wrote, using tensor notation. I was just having a hard time even getting started with that whole idea of taking a magnitude in tensor notation.

 

[Dustinsfl]

Well actually the goal of the question is to prove that expression you just wrote, using tensor notation. I was just having a hard time even getting started with that whole idea of taking a magnitude in tensor notation.

Here is a homework of mine from Continuum Mechs with Tensor problems worked out:
http://ubuntuone.com/4qjtmJJmCXpewCPoKNJhXf

 

[skate_nerd]

That really helps a lot. Definitely going to bookmark that, for future reference :D