The discussion centers on Einstein's summation convention in tensor calculus, specifically addressing the implications of using indices. According to the convention, an index appearing twice indicates summation, while appearing more than twice renders the expression meaningless. The ambiguity arises from the need to distinguish between superscript and subscript indices, which define the type of tensor (covariant, contravariant, or mixed). The conversation emphasizes that if an expression contains three identical indices without a corresponding superscript, it is considered ambiguous and should be clarified by the author.
PREREQUISITESStudents and professionals in mathematics, physics, and engineering who are learning or applying tensor calculus, particularly those interested in the nuances of tensor notation and summation conventions.
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?