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.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?
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.
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.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?
Einstein's summation convention is a mathematical notation used in tensor analysis and other areas of mathematics. It involves summing over repeated indices in a formula, making calculations involving tensors and matrices more concise and easily readable.
In Einstein's summation convention, when an index appears twice in a term, once as a subscript and once as a superscript, it is implied that the term should be summed over all possible values of that index. This eliminates the need for explicit summation symbols, making equations more compact.
The main benefit of using Einstein's summation convention is that it simplifies and streamlines mathematical expressions, making them easier to read and understand. It also allows for faster calculations and reduces the chances of errors in calculations involving tensors and matrices.
One limitation of using Einstein's summation convention is that it is not suitable for all types of mathematical equations. It is primarily used in tensor analysis and other areas of mathematics where there are repeated indices. Additionally, it may not be as intuitive for those who are not familiar with the notation.
Einstein's summation convention is named after Albert Einstein, as he was the first to use this notation in his work on general relativity. However, it is important to note that the convention was not created by Einstein himself, but rather became popular because of his use of it in his theories.