I have recently been introduced to two rather abstract symbols. I consider them abstract because I have not seem them in my undergrad studies. These two symbols are the Kronicker Delta and the Levi-Civita Symbol.
I understand that these two symbols are essentially multipliers. I understand them best when they are used to calculate the dot product and the cross product of a vector. Thus far I have seen examples of these symboles notated as follows:
Delta Kronicker: dij, dji Levi-Civita: Eijk, Ejki, Ekji, etc. where d equals delta, and E equals epsilon.
My confusion begins with the following:
"The inverse of
di = (Eijk)Tjk
Where di is a "Dual Vector", and Tjk is a Tensor.
is found by multiplying both sides by Eilm, that is,
(Eilm)di = (Eilm)(Eijk)(Tjk)
I understand what Tjk is (T11, T12, T13, T21, T21, ... T33). However Eilm is new to me and I do not know what sets Eijk apart from Eilm. I am familiar with i, j, k because undergrad courses always used them as the x, y, and z axis. Is this why we start with Eijk? If so, what is Eilm? Eilm seems to be quoted in my text often, yet what is the lm in Eilm? for example:
(Eijk)(Eilm) = (djl)(dkm) - (djm)(dkl)
I can see further ahead in my text, Eipq, etc is used. Are these leters just arbitraty indices? If so, is Eilm just next in line?
Perhaps you can refer me to an online article or journal that may be able to explain these differences a bit better?