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According to a version of Einstein summation convention, an expression with a repeated dummy index that appears as both a prefix and a suffix is understood to be a summation over that index. Under this convention, the prefix is not an exponent. To distinguish a prefix from an exponent, a bracket has to be used. For example, the traditional expression $\sum_i a_ix_i{}^2$ has to be written as $a_i(x^i)^2$ under the convention.

Now I have a question. How do we wite an expression with an exponent being summed under the convention? For example, how do we write the traditional expression

$\sum_{i = 1}^n a_ix^i = a_1x^1 + \ldots + a_nx^n$

under the convention? Can we write it in the following way?

$a_i(x)^i$

Ka Fat Chow
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    Usually you don't. The Einstein summation is classically only used for the Ricci tensor calculus. – Lutz Lehmann Sep 30 '22 at 04:32
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    The Einstein notation is good because it's the same for all vector spaces - you sum over a basis. What you wrote is something else, and it would be a bad convention because it's just confusing given that Einstein sum exists. – Daniel Teixeira Sep 30 '22 at 04:43
  • Einstein notation is used to write a tensor with respect to a basis that is naturally induced by a basis of the underlying vector space. The indices label the basis elements. You would always label them with an index. – Deane Sep 30 '22 at 12:32
  • A coefficient of a Taylor series has powers but each term can be viewed as a symmetric tensor. – Deane Sep 30 '22 at 12:34
  • In your example each term is a 1-dimensional tensor. Using Einstein notation still works fine. The indices just run from $1$ to $1$. – Deane Sep 30 '22 at 12:36

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