- #1

Vanilla Gorilla

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- TL;DR Summary
- 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.

$$(a_i x_i)_{e}= (\sum_{i=1}^3 a_i x_i)_r=(a_1 x_1+a_2 x_2+a_3 x_3)_r$$; 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.