Einstein summation notation, ambiguity?

In summary, there is a way to distinguish between summing in the domain of f and summing in the range using Einstein's summation notation. The latter can be written explicitly, overriding the convention, and involves using the Kronecker delta. This insight allows for clearer representation of the desired summation.
  • #1
Hiero
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If I see ##f(x_ie_i)## I assume it means ##f(\Sigma x_ie_i)## (summing in the domain of f) but what if I instead wanted to write ##\Sigma f(x_ie_i)## (summing in the range)?

Is there a way to distinguish between these in Einstein’s summation notation?
 
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  • #2
This is an odd case, but I think if you want the latter, in a context where Einstein summation is implied, you have to write it explicitly, overriding the convention. Only first case is handled by the summation convention.
 
  • #3
Hiero said:
Is there a way to distinguish between these in Einstein’s summation notation?
So ##f(\Sigma x^{\mu}e_{\mu})## is ##f(x^{\mu}e_{\mu})## and ##\Sigma f(x^{\mu}e_{\mu})## is ##f(x^{\mu}e_{\nu})\delta^{\nu}_{\mu}##
 
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Likes andresB, sophiecentaur, PeroK and 1 other person
  • #4
Dale said:
So ##f(\Sigma x^{\mu}e_{\mu})## is ##f(x^{\mu}e_{\mu})## and ##\Sigma f(x^{\mu}e_{\mu})## is ##f(x^{\mu}e_{\nu})\delta^{\nu}_{\mu}##
Oh wow, that’s clever! Thanks for the insight.
 
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Likes Dale

Related to Einstein summation notation, ambiguity?

1. What is Einstein summation notation?

Einstein summation notation, also known as Einstein notation or tensor notation, is a mathematical convention used to simplify the representation of equations involving tensors and matrices. It involves using Greek indices to represent different components of a tensor or matrix, and summation symbols to indicate that all possible combinations of these components should be summed up.

2. How is Einstein summation notation used?

Einstein summation notation is used to express complex mathematical equations in a concise and efficient manner. It is commonly used in fields such as physics, engineering, and mathematics to represent equations involving vectors, matrices, and tensors.

3. What is the ambiguity in Einstein summation notation?

The ambiguity in Einstein summation notation refers to the fact that there can be multiple ways to interpret the notation, depending on the context in which it is used. This can lead to confusion and errors if not used correctly.

4. How can ambiguity in Einstein summation notation be resolved?

To resolve ambiguity in Einstein summation notation, it is important to clearly define the indices and their ranges, as well as the order in which the summation is performed. It is also helpful to provide a diagram or explanation of the underlying mathematical concept to aid in understanding.

5. What are some common mistakes made with Einstein summation notation?

Some common mistakes made with Einstein summation notation include using the wrong indices, forgetting to include all possible combinations in the summation, and misinterpreting the order of operations. It is important to carefully check and double-check the notation to avoid these errors.

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