High School Beginner Einstein Notation Question On Summation In Regards To Index

[Vanilla Gorilla]

TL;DR

More generally, in Einstein Notation, do we ONLY sum over the dummy indexes, which constitute ALL indexes that occur twice or more in a single term?

So, I have recently been trying to learn how to work with tensors. In doing this, I have come across Einstein Notation. Below is my question.

$$(a_i x_i)_{e}= (\sum_{i=1}^3 a_i x_i)_r=(a_1 x_1+a_2 x_2+a_3 x_3)_r$$; note that the following expression is in three dimensions, and I use the subscripts "e" to denote when I am using Einstein Notation and "r" to denote when I am using 'regular' notation, which I am more comfortable with.
My question is, are these terms - $a_1 x_1+a_2 x_2+a_3 x_3$ - implied to be summed over? I believe that the answer is no since there is no dummy index that we would sum over, but I'm not 100% sure;
If I'm correct, and we don't sum, would that mean that, $a_1 x_1$, for example, just implies regular multiplication here?

$$w V^r$$
Likewise, $w$ would not be summed over here by the same logic (It has no index, and can thus not be a dummy indexed term).

More generally, in Einstein Notation, do we ONLY sum over the dummy indexes, which constitute ALL indexes that occur twice or more in a single term?

P.S., I'm not always great at articulating my thoughts, so my apologies if this question isn't clear.

 

[Ibix]

I think you have it right. For example $aM_{ijk}x_ix_k$ expands to $\sum_i\sum_kaM_{ijk}x_ix_k$. We sum over $i$ and $k$ because they appear exactly twice, and we don't sum over $j$ because it only appears once. You can take $a$ outside the summation signs if you like, since it is a common factor in the terms of the sum.

If you are using Einstein notation then triply (or more) repeated indices are illegal - $u_iv_iw_i$ is not allowed. If you need it for some reason then you have to write out the sum explicitly.

If the indices have been replaced by specific values like 1, 2, 3, then no sum is implied. You do sometimes see some slight abuse of the notation where (e.g.) $x$, $y$, and $z$ are taken to be specific values when they are the names of coordinates - so $a_xb_x$ might be interpreted as the prodict of the $x$ components of $a$ and $b$, so you have to pay attention.

 

[Vanilla Gorilla]

This makes sense and cleared up my confusion! Thank you so much