I'm studying for a vector calculus exam and I'm trying to discern if there are multiple ways to express the following equation in Einstein summation convention(a,b,u,v are vectors in $\mathbb{R^3}$):
$$ \mathbf{u} + (\mathbf{a} · \mathbf{b})\mathbf{v} = (\mathbf{a}·\mathbf{a})(\mathbf{b} · \mathbf{v})\mathbf{a} $$
The answer given in the notes is:
$$ u_i + a_jb_jv_i = a_ja_jb_kv_k a_i $$
I'm curious if this is equivalent:
$$ u_i + a_jb_jv_i = a_ka_kb_jv_j a_i $$
Are the indexes j and k are interchangeable on the right hand side of the equation? Or is it necessary that the indexes for (a · b) and (a · a) are the same?
Consider un-Einsteining the notation for a second, just to make it a bit clearer: $$ u_i + \sum_{j = 1}^3a_jb_jv_i = \sum_{j = 1}^3\sum_{k = 1}^3a_ja_jb_kv_k a_i\\ \text{versus}\\ u_i + \sum_{j = 1}^3a_jb_jv_i = \sum_{k = 1}^3\sum_{j = 1}^3a_ka_kb_jv_j a_i $$ The $k$ and the $j$ on the right-hand side don't touch the other terms. The actual name you choose for each of them does not matter at all for the rest of the equation, because it doesn't "leak" outside of their respective $\sum$. You could just as well have written $$ u_i + \sum_{j = 1}^3a_jb_jv_i = \sum_{\dagger = 0}^3\sum_{わ=0}^3a_\dagger a_\dagger b_わv_わ a_i\\ $$ Of course, this only applies for the names of indices that are summed over / contracted within a single term (including the $j$ on the left-hand side). The $i$ must be chosen to be consistent across the whole equation and not collide with any of the others. Although, there is no reason it has to be $i$ specifiacally.