Question regarding Einstein's summation convention

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    Convention Summation
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Discussion Overview

The discussion revolves around the Einstein summation convention in tensor calculus, specifically addressing the implications of index appearances in tensor expressions. Participants explore why an index appearing more than twice is considered meaningless within this convention and the ambiguity that arises from superscript and subscript indices.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that according to the summation convention, an index appearing twice indicates summation, while appearing more than twice renders the expression meaningless.
  • Others argue that if a capital sigma is used, the expression can be meaningful regardless of how many times an index appears.
  • Concerns are raised about ambiguity in summation when both superscript and subscript indices are present, questioning how to determine which indices to sum over.
  • Participants discuss the case of three identical subscript indices without any superscript index, questioning whether such an expression is meaningless or if it is only meaningless under the summation convention.
  • One participant suggests that tensor notation is meant to simplify evaluations and that using three identical indices can lead to ambiguity, indicating that the author should clarify the expression.
  • A suggestion is made to consult external resources, such as a Wikipedia article on Einstein summation, to gain further clarity on the topic.

Areas of Agreement / Disagreement

Participants express differing views on the implications of multiple index appearances and the clarity of tensor notation. There is no consensus on whether expressions with three identical indices are inherently meaningless or if the context of the summation convention is what renders them so.

Contextual Notes

Limitations include the potential ambiguity in tensor notation when multiple indices are used and the reliance on the summation convention, which may not apply universally in all contexts.

rohitgupta
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I have just started on a course in Tensor calculus and I'm absolutely new to it, so I read that according to the summation convention, if an index appears twice, it means that the expression is summed over that index, but if it appears more than twice then the expression is meaningless. I want to know why it is meaningless? I mean why can't an index appear thrice?
 
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rohitgupta said:
I have just started on a course in Tensor calculus and I'm absolutely new to it, so I read that according to the summation convention, if an index appears twice, it means that the expression is summed over that index, but if it appears more than twice then the expression is meaningless. I want to know why it is meaningless? I mean why can't an index appear thrice?
Since it is a convention, appearing thrice is meaningless by that convention. If you put a capital sigma in front, then it is meaningful no matter how many times a particular index appears.
 
Usually in Einstein summation its a superscript index and a subscript index of the letter that is summed so you can imagine the ambiguity. how would you know which super and sub to sum over? the superscript and subscripts define the kind of tensor whether covariant contrvarient or mixed.
 
jedishrfu said:
Usually in Einstein summation its a superscript index and a subscript index of the letter that is summed so you can imagine the ambiguity. how would you know which super and sub to sum over? the superscript and subscripts define the kind of tensor whether covariant contrvarient or mixed.

What if there are 3 subscript indices in the expression, all of which are same and there is no superscript index.

I mean is the expression meaningless or is it only meaningless in the summation convention?
 
rohitgupta said:
What if there are 3 subscript indices in the expression, all of which are same and there is no superscript index.

I mean is the expression meaningless or is it only meaningless in the summation convention?
I wouldn't apply the summation convention in that case. Tensor notation is used to abbreviate how to evaluate it. Sometimes people will write Tii=1 (both ii as subscripts) to mean the tensor elements T11, T22, T33, in 3-space and not T11 + t22 + T33.

Remember if its written with 3 indies its ambiguous and the author should fix the equation not have the reader guess.
 
probably its best that you read the wikipedia article on Einstein summation. it seems you are delving too deeply into the fact that they simply remove the Ʃ from the equation. SO the author would use that convention and realizing that there could be an issue would not confuse things using the same index letter again elsewhere. Also notice that the convention is one upper index and one lower index.

The article shows examples at the end and even some where there's two upper indexes that aren't summed over.
 
Your last reply clears it. I get it now. Thank You.
 

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