# Question regarding Einstein's summation convention!

## Main Question or Discussion Point

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?

## Answers and Replies

mathman
Science Advisor
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.

jedishrfu
Mentor
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.

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?

jedishrfu
Mentor
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.

jedishrfu
Mentor
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 theres two upper indexes that aren't summed over.

Your last reply clears it. I get it now. Thank You.