# Common Einstein Notation Identities

I find myself regularly looking up common vector identities in index-tensor notation like the following simple examples

$(u\times v)_i = \epsilon_{ijk} u_j v_k$ (in 3-space)

or

$u\cdot v = u_iv_i$

$(MN)_{ij} = M_{ik}N_{kj}$

(with the implied summation over repeated indices - abuse of Einstein notation where the covariant/contravariant distinction is not important)

These are easy to remember, but other - more complex - vector/matrix expressions either require working them out by hand, or hunting them down

Is there a reference available in "cheat sheet" form that lists a good amount of these identities concisely? Something like a table of integrals but for common vector memes written in index notation.

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Don't you have any examples of things that are not easy to remember which you'd want in a cheat sheet? What have you needed to work out by hand and/or hunt down recently? – Henning Makholm Oct 5 '11 at 22:03
Without meaning to be unkind: you might want to brush up on your linear algebra. The formulae for the dot product and for matrix multiplications follow immediately from the definitions for matrices. Aside from the cross product, which more or less motivates the definition of the Christoffel symbol, it's not clear how many such formulae exist which don't just amount to translating basic facts of linear algebra into the new notation. – Niel de Beaudrap Oct 5 '11 at 22:08
@NieldeBeaudrap they weren't meant to be complex examples. For instance working out $A(u)\times A(v)$ where $R$ is 3-by-3 matrix takes a few moments and is something I've encountered a few times. – crasic Oct 5 '11 at 22:11
You mean like $\epsilon_{ijk} A^j_a u^a A^k_b v^b$? The conversion to Einstein notation can be done just by composing the translation rules and fishing for new indices as needed. Do you mean the conversion back from Einstein notation, or just how you would chain together the translation? – Niel de Beaudrap Oct 5 '11 at 22:14