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transpose in std notation

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A short working note by Phil (PhL, dated 4.9.12) weighing whether to retain the transpose T in standard index notation or drop it from his tensor document. He defines (A^T)_ab as A_ba, checks consistency with developmental notation, and worries about orthogonality of S holding in standard but not developmental notation. He also cites web sources and a Plan A that converts A A^T = 1 to standard notation. A header says he decided to keep T and heavily edit the section.

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The Transpose T in Standard Notation (keep) PhL 4.9.12 [ Well, I decided to retain T in standard notation, and this led to a severe edit of that section. All below is OK, and I show the thin support I found on the web for my T approach. ] I am leaning strongly toward removing this from tensor doc and NEVER using a transpose symbol in standard notation. Weinberg has no need for such a symbol. Most of my stuff is now in Section 7 (i) and I have never liked it. Let's just see where ELSE I have used it. Search on capital T. Section 7 (r) has some. You can just search for "RT" to find things. Anything prior to Section 7 will always use dev not, so that should not be a problem. It is in and after Section 7, and the appendices where I am concerned. Really there are no references outside Section 7. How about the Appendices? Nothing really. I could just make a rule: when RT appears, developmental notation is implied. So what is WRONG with my transpose idea? My presentation is in Section 7 (i) item 8. The discussion concerning "all same tilt" matrix multiplication seems OK to me. Discussion: I start saying that (AT)ab ≡ Aba which seems awfully reasonable. If you do this, and if Aba is a rank-2 tensor, then (AT)ab is also a rank-2 tensor, and all the rest correctly follows. Is this "consistent" with the dev notation? Aab = ATba → Aab = (AT)ba That much seems OK. What I really don't like is the conclusion that S is real orthogonal for any underlying F and how this is true in std notation but NOT in dev notation! What does the web say on this topic? Here is one PDF: You cannot even tell which index is first and which second, so I cannot trust this guy for anything. A lady from Utah does it my way She never does anything with the transpose object, does not comment on the seeming paradoxes. Plan A. Suppose A is a true tensor. Then always start in dev not AAT = 1 says Aij(AT)jk = δik which is the same as AijAkj = δik To convert this to std not, I would take Aij to be a contravariant tensor, so it would become AijAkj = δik This is NOT a tensor equation because the j index is not tilted.