I'm studying the component representation of tensor algebra alone. There is a exercise question but I cannot solve it, cannot deduce answer from the text. (text is concise, I think it assumes a bit of familiarity with the knowledge) (a) Convert the following expressions and equations into geometric, index-free notation: AαBγβ ; AαBγβ ; Sαβγ=Sγβα ; AαBβ=AαBβgαβ In this problem, I can't see any difference between first two expressions except for the index position, and my only solution for the expression of index position is using metric tensor g, which I think is included in slot-naming notation. Is "index-free" notation can express the difference? Other expressions are similarly confusing for me. (b) Convert T(_,S(R(C,_),_),_) into slot-naming index notation. I think this notation would be not so universal notation. These problem are from http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1202.1.K.pdf (Ex 2.7) and the help of anyone who is familiar with the notation would be appreciated.