Graduate Upper indices and lower indices in Einstein notation

[GGGGc]

I have read some text about defining the cross product. It can be defined by both a x b = epsilon_(ijk) a^j b^k e-hat^i and a x b = epsilon^(ijk) a_i b_j e-hat^k
why the a and b have opposite indice positions with the epsilon? How to understand that physically?

 

[GGGGc]

This is the text i read

 

[PeroK]

Try this to get you started:

https://en.wikipedia.org/wiki/Einstein_notation

 

[GGGGc]

Try this to get you started:

https://en.wikipedia.org/wiki/Einstein_notation

I've already checked that, but I'm still confused about when to use upper or lower indices, do you have some examples? Thanks for answering my question!

 

[PeroK]

I've already checked that, but I'm still confused about when to use upper or lower indices, do you have some examples? Thanks for answering my question!

You really need a textbook or lecture notes, as there is a lot to say on this. The basic idea comes from the notaion of a dual vector space. In Euclidean space, the dual space can be associated directly with the original space, so there is no need for a distinction. And, generally a vector is written as a sum of its components in any coordinate basis:
$$\mathbf a = \sum_{i = 1}^n a_i\mathbf e_i$$Where $\mathbf e_i$ are the basis vectors. When we come to curved spacetime, the dual space can no longer be directly associated with the original space. Therefore, we have a coordinate basis and a dual basis and a vector is expressed in terms of the basis and a dual vector in terms of the dual basis. (Note that this concept extends to tensors of any rank.) The einstein notation not only dropped the summation symbol, but used upper indices to represent components of a vector and lower indices to represent the basis vectors. This is because the components obey the contravariant transformation law and the basis vectors obey the covariant transformation law (you better check I've got that the right way round!). So, we write:
$$\mathbf a = a^{\alpha}\mathbf e_{\alpha}$$Conversely, the components of a dual vector (also known as a one-form) obey the covariant transformation law and the basis dual vectors obey the contravariant law. So, for a dual vector, we write:$$\mathbf w = w_{\alpha}\mathbf \theta^{\alpha}$$Where $\mathbf \theta^{\alpha}$ are the basis dual vectors.

That should get you started, but whatever you are studying should go into the Einstein notation in sufficient depth.

 

[GGGGc]

Thank you so much for your explanation! I'm quite clear now!