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Writing a simple sum with einstein notation

Physics Asked by l3utterfly on December 11, 2020

Is there any shorthand using the Einstein notation to write a simple sum?

$X = X_1 + X_2 + X_3 ….$

Can I write: $X = X_i$ ?

I don’t think so because the definition states that only indices that appear twice is summed over. What is the correct way to write a simple sum like the example above?

2 Answers

The Einstein summation notation includes repeated indices (i.e. summation notation). For vectors (and tensors), $X$ would usually mean the full object, $X = X^{i} e_{i} = X^{0}e_{0} + X^{1}e_{1}...$ etc. The $X^i=X^i (x)$ are the components of the vector and $e_i$ is some basis. The $i$ deonotes the components, and this is summed in the above equation. But we can talk just about the components $X^i$, which means $(X^0, X^1, ... X^n)$.

So yes, you can write $X^i$, but it doesn't mean what you've written down. This is just four-vector notation, but not specifically Einstein summation notation. The sum you wrote is more like the Euclidean norm, which as the other comment says, it's easiest notated as $sum_{i}^{n} X^i $.

Answered by Eletie on December 11, 2020

Such a sum is, as far as I know, not easily expressible in Einstein notation - you could probably write it in a convoluted way, but that would not really be simpler or easier to understand than $X = sum_i X_i$.

The summation convention is not a shorthand for any sum, but only for certain kinds of sums which arise in linear algebra, such as scalar products, matrix multiplication and so on. In formal terms, the sums at hand are those corresponding to the application of (multi)linear maps between a vector space, its dual, and powers of these; or to the coordinate representation of vectors and tensors. Some examples are given in the Wikipedia article for the convention.

This is the reason for the requirement of two equal indices.

Answered by Jacopo Tissino on December 11, 2020

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