I am going to show how to prove the following equality using Summation Notation, Kronecker Delta’s, and Levi-Civita Notation: . Where , , , and are three dimensional vectors. Throughout the proof, I will explain what each of these symbols means and why I am using them. I am assuming that the reader is familiar with the idea of dot products and cross products.
I begin by letting and . and are still vectors and so I can convert their dot product into summation notation. This will look like . What this notation means is that for each from 1 to 3 (1 representing the x-component, 2 representing the y-component, etc.), I am multiplying respective components of and and then adding the product of the next components until I have done this with all 3 components. Since I am only dealing with vectors containing 3 components I will drop the term and just write , and I will also drop the upper summation and leave it blank. I can substitute back in the respective cross products for and . This will look like . For this particular proof, I will only be dealing with the x (i = 1) component of the vectors to show the identity. I will now convert the cross products into similar summations using Levi-Civita notation.This will look like . In this notation, the first notation is taking care of . The next four summations are evaluating each cross product, and the epsilon notation is what is known as Levi-Civita Notation, which is essentially a piece-wise function that assigns a 1, -1, or 0 to each component in the summation depending on the permutation of sub indices of . For example, the cross product of two unit vectors and produces a new vector, namely which is perpendicular to both. If I wanted to do this same cross product but use Levi-Civita notation and summation notation, I would use the formula . Two substitutions must now be made. The first is that . This comes from making permutations in the indices of the epsilon. Now, a second substitution must be made which relates a product of two epsilons to a difference in Kronecker Deltas. A Kronecker Delta is essentially a piecewise function that assigns a or a to a dot product between two vectors depending on the subindices of the two vectors and therefore the sub indices of the Kronecker Delta. This identity is . After I make this substitution, the Sum can be reorganized so that it looks as follows: . After factoring in the vectors to the difference in deltas I can deal with the first sum of quantities which looks as follows: . The only case where the Kronecker Deltas don’t equal zero is when two of their indices are the same. When they are not the same, the Deltas are equal to zero and therefore don’t contribute anything to the summation. With this in mind, I can set which will cause the second delta to equal 1 and the fourth summation to be a sum over k. Since this summation already exists, the fourth sum can just be dropped. I am then left with . Now, I can set which will lead to the final delta equating to 1 and the third summation being dropped. Finally, I am left with which can be reorganized to look as follows: . I now recognize that each of the quantities in parentheses are dot products, summed over two different indices, which is equivalent to . If I were to then follow similar steps to the second product of Deltas in the second substitution that i made above I would find that I would eventually obtain .