Please me to understand Einstein notation and permutations

[Tiggy]

Hey,
Can anyone help me to understand einstein notation and permutations? I have a book, but it's not very clear. I really don't understand how you can write A_ij = e_ijk a_k out as a matrix? To start with I understand that a matrix can be represented as A_ij where i is the row and j is the column. However, I don't understand as soon as k and the kronecker delta become involved! Anyone know of any good way to explain this or books or websites??

Thank you!

 

[robphy]

http://en.wikipedia.org/wiki/Levi-Civita_symbol
http://mathworld.wolfram.com/PermutationSymbol.html
http://planetmath.org/encyclopedia/LeviCivitaPermutationSymbol3.html

(I assume that you already understand the summation convention.)

 

[tacman]

Sure, I'd be happy to help you understand Einstein notation and permutations. Einstein notation, also known as index notation or tensor notation, is a way of writing mathematical equations involving tensors (objects that have both magnitude and direction, such as vectors and matrices). It simplifies the notation and makes it easier to perform calculations involving tensors. In Einstein notation, repeated indices in an equation indicate summation over those indices, similar to the summation convention in traditional algebraic notation. This allows for more compact and concise writing of equations.

As for the permutation aspect, permutations refer to the rearrangement of the indices in a tensor. In Einstein notation, this is denoted by the kronecker delta, which is a symbol that represents a matrix with ones on the main diagonal and zeros everywhere else. This allows for the rearrangement of indices without changing the value of the equation. For example, A_ij = A_ji, where the kronecker delta ensures that the two sides of the equation are equal.

To write A_ij = e_ijk a_k as a matrix, you would use the kronecker delta to rearrange the indices. For example, if we have a 3x3 matrix A, we can write it in Einstein notation as A_ij, where i and j range from 1 to 3. The kronecker delta allows us to rearrange the indices to A_ji, which would result in the same matrix but with the rows and columns switched. Similarly, the e_ijk term represents a tensor with three indices, and the kronecker delta allows us to rearrange the indices to e_jik, which again results in the same tensor but with the indices rearranged.

I hope this explanation helps to clarify things for you. As for additional resources, you may want to check out textbooks on linear algebra or tensor analysis, as well as online tutorials or videos on Einstein notation. Practice and exposure to different examples will also aid in your understanding. Let me know if you have any further questions or need clarification. Good luck!