Is Einstein notation universally applicable?

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It gets surprisingly monotonous to write the sum symbol repeatedly, so I'm wondering if one can just swap in Einstein notation at will to avoid the monotony, assuming that the symbol that's being summed over appears exactly twice?

For example, the wiki definition of $\nabla$ uses the sum symbol:

$\nabla = \sum_{i=1}^n \vec e_i {\partial \over \partial x_i} $

Could one simply write

$\nabla = \vec e_i {\partial \over \partial x_i} $

instead, and if so, would it need to be prefaced with "Using Einstein summation notation... " or is even that necessary?

If this question is more appropriate for physics.se, I'll post it there. Math.se has more hits on the search "Einstein notation," so I thought I'd ask here first.

Thank you for any help.

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4 Answers

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I agree with Don’s comment this is confusing and unwelcome outside areas of physics where it’s common. I disagree that it makes a lot of sense.

If $a_i\vec{e}_i$ is a sum, how would you talk about an particular but arbitrary $a_i\vec{e}_i$ when you needed to? Maybe physicists never have to do that?

If you multiply two sums together that both use the index $i$, you have to change one of the letters if you aren’t explicit about the sum. This is likely confusing to a mathematics student. Why do you sometimes have to change letters when you multiply things, but at other times can’t?

Another problem with using this generally: mathematics often includes various different index sets in single expressions: $\sum_{i=0}^n\sum_{j=1}^{i-1} a_{ij}$. It’s not easy to see when you can and when you can’t write such expressions in Einstein notation.

There are a lot of confusing consequences of this notation outside a narrow context.

[Disclaimer: I was a physics major for a while, and this notation (among other things) led me to study mathematics.]

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My feeling is that this is used much more broadly than in physics and it will be read as intended by any audience that's learned Einstein notation, so I use it readily, and will only mention what it is if I suspect the audience might not know it. For example, in a math.se answer, if the questioner didn't use the notation themselves, I try not to assume they know it already. I still use it, but briefly name it (once per document ought to do, especially with the hyperlink I've included above, even though you didn't need an explanation).

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I would just like to add few comments to the answers posted here.

First of all, I am afraid there is nothing universally accepted, especially when regarding notation. It is true however that Einstein notation alleys in almosy every scientific paper related to General Relativity and is also common in theoretical physics. In maths is more difficult to see it, but I do think most of the geometers are aware of its existence (although they might not like it).

Regarding how useful/useless it is, I would say it depends. It depends on the equations you need to write. If in your expressions repeated symbols are always adding then go ahead. You can use it even if that's not the case for all your expressions, you can just add a warning remark between brackets "(no sum)" or "(no sum over $i$)" before displaying the exceptional ones.

For the other pathological case mentioned in a comment, I suggest two possible solutions: either the indices are clearly different (for instance of you are considering a tensor product of different spaces it is common to denote the first coordinates by $x_i$ for example and use $y_\alpha$ for the others and the reader knows the ranges of each indexing set) or in that particular case the sum is explicitly written.

Summarising, that would be the way I suggest that you solve the possible issues:

Throughout the text, Einstein notation will be adopted when there will be no risk of confusion. In such cases, and if nothong else is stated, he reader is intended to interpret repeated indices affected by a sum symbol, ranging over the corresponding indexing sets. In the cases where there may be a possible misunderstanding, sums will be explicitly written.

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I'm studying mathematics and my favourite area of research is Differential and Riemann Geometry among Tensor and Exterior Algebra. I must say that in these fields, even though they can be heavily applied to physics, it's very common and actually nice to represent things using Einstein Notation and Tensorial Notation. Let's take a look at this example in Vector Calculus:

There's an identity that stands for (I'm not explaining here what it means, I'm only concerned about the notation that can be used in a pure maths scenario):$$A \times (\nabla \times B) = \nabla_B\langle{A,B}\rangle - \langle{A, \nabla}\rangle B$$

You may ask, what the **** is this? Don't worry, so do I and all mathematicians on earth, what is $\nabla_B$ for example? No way it can be understood in a regular vectorial notation used in Vector Calculus. Again, I'm not explaining the meaning of it but take a look how I can rewrite the identity above using only einstein summation and tensor notation:

$$\epsilon^{rsi}A_s\epsilon_{ijk}\nabla^jB^k=(\delta^{r}_{j}\delta^{s}_{k} - \delta^{s}_{j}\delta^{r}_{k})A_s\nabla^jB^k=A_s\nabla^rB^s - A_s\nabla^sB^r$$

Ok, being quiet honest if you're not familiar with tensor and einstein notation the latter will make you tear apart your sheets, but trust me, once you learn these notations the latter identity will make you understand the first identity (which are actually the same) and you will see that it looks PRETTY MUCH harder than it actually is.

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