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In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?

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Not that I know of - I would probably just say "$k^{th}$ upper/lower index". Personally, if I had to deal with a tensor with more than two indices, I wouldn't write its components unless I absolutely had to :) – Neal Jan 22 '13 at 15:21
-1 Tensor is not a matrix, so the question doesn't really make sense. – Marek Jan 22 '13 at 15:21
I respectfully disagree with the downvote. We could give names to the indices, even in the nonsquare case, I don't see anything senseless with that. Sometimes, giving names does help intuition. – Giuseppe Negro Jan 22 '13 at 15:35
@Marek I disagree with you. Often, students of physics or engineering learn that tensors are generalized matrices which transform according to certain rules. From this perspective, conceptualizing the "first" two indices as "row" and "column" makes good sense, and is not IMO worthy of a downvote. – Neal Jan 22 '13 at 15:43
@Neal: I'm facing a programming problem so I have to use a name for the total number of "things" analogous to rows and columns. Currently I call it matrices. – August Karlstrom Jan 22 '13 at 15:49
up vote 3 down vote accepted

A common name after row and column is tube, according to Google. More generally, one speaks of fibers of a tensor, enumerating them by modes. The illustration below is from the book Nonnegative Matrix and Tensor Factorizations by Andrzej Cichocki, Rafal Zdunek, Anh Huy Phan, and Shun-ichi Amari, published by John Wiley & Sons in 2009 (I surely hope reproducing one illustration is fair use).


(And +1 to the question; it received one of the more unreasonable downvotes I've seen around here.)

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