# Learning Index Notation: Tensor/Cross Product Confusion

• Incand
In summary: As for the swapping of terms, it doesn't matter as long as you keep track of the signs and permuted indices as shown in the identity. You could swap two terms at a time or use sorting algorithms, as long as the result follows the identity.
Incand
I just started learning index notation and I'm having some trouble understand what I'm allowed to do with it.
For example we can write the ##\vec A \times (\vec B \times \vec C)## as ##\varepsilon_{ijk}A_j\varepsilon_{klm}B_lC_m##. I understand that ##(\vec B \times \vec C) = \varepsilon_{klm}B_lC_m##
but why am I allowed to just throw in another vector and another ##\varepsilon##-tensor to get another cross product? How would this write if i write out the summation symbols, do I sum over everything or does the order matter? Like this?
##\sum_{k=1}^3\sum_{j=1}^3\sum_{l=1}^3\sum_{m=1}^3 \varepsilon_{ijk}A_j\varepsilon_{klm}B_lC_m##
Does it matter in what order i sum this up? why not? The "tripple cross product" isn't commutative is it, so the order should matter? Why am I allowed to use the identity ##\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}## here?

As you can se I'm really confused about pretty much everything having just started out the subject and I can't make much sense of the book I'm using sadly so some help clarifying it for me would be awesome!

The only way you're going to understand this is to expand the tensor notation out, simplify the expression (ie remove canceled terms) and compare the result to a known vector identity.

https://en.wikipedia.org/wiki/Levi-Civita_symbol

Try to prove the ##\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}##

and in your expansions look for terms with permuted indices and recall that ##\varepsilon_{123} = \varepsilon_{231} = \varepsilon_{312}## and that ##\varepsilon_{123} = -\varepsilon_{213}##... and lastly when any two indices are equal as for example ##\varepsilon_{iik} = 0##.

This will remove some terms and eventually you will be left with a componentized version of the identity.

This works for the vector identities too.

Here's a youtube video discussion on it:

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Incand
Thanks for responding, that video were very usefull! Especially the end when he proved those statements .When I did it myself I wrote out a list of all the possibilities (sometimes eliminating some thanks to symmetry) and then verified them. His way of doing it is a lot cleaner (and faster).

I understand the identity ##\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}## and I'm able to prove it (in fact the book I'm using does). What I was mainly confused about was how I was allowed to change order of the terms.
Mainly how I could write
##\varepsilon_{ijk}A_j\varepsilon_{klm}B_lC_m = \varepsilon_{ijk}\varepsilon_{klm}A_jB_lC_m##
Is this because everything here is a scalar and therefore I'm allowed to change the order? I guess that's the reason index notation is so powerful.

In that video he seems to make a point out of only swapping two terms in every step (when showing the identities in the last part) but that is just to make it easier to follow I assume? Since if you're allowed to swap any two elements however you want, you could sort the terms in any way you want (thinking about sorting algorithms here).

Yes, everything is a scalar and so the ordinary laws of algebra apply.

Incand

## 1. What is index notation and how is it used in science?

Index notation is a way of writing and manipulating mathematical expressions that involve multiple variables. In science, it is commonly used to represent physical quantities, such as vectors and tensors, and to perform operations on these quantities.

## 2. What is the difference between a tensor and a cross product?

A tensor is a mathematical object that represents a physical quantity that has both magnitude and direction. A cross product, on the other hand, is a mathematical operation that takes two vectors and produces a new vector that is perpendicular to both of the original vectors.

## 3. How do I perform a tensor product and a cross product using index notation?

To perform a tensor product, you can use the Einstein summation convention, where repeated indices are summed over. For a cross product, you can use the Levi-Civita symbol, which has values of -1, 0, or 1 depending on the order of the indices.

## 4. What are some common mistakes to avoid when using index notation for tensors and cross products?

One common mistake is confusing the order of the indices in a cross product, which can result in an incorrect direction for the resulting vector. Another mistake is forgetting to use the Einstein summation convention, which can lead to incorrect calculations.

## 5. How can I practice and improve my understanding of index notation for tensors and cross products?

One way to practice is by working through practice problems and exercises, either on your own or with a study group. It can also be helpful to review and understand the underlying mathematical principles behind index notation and how it relates to physical quantities. Additionally, seeking help and clarification from a teacher or mentor can also aid in improving your understanding.

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