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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:

A

^{α}B

_{γβ};

A

_{α}B

_{γ}

^{β};

S

_{α}

^{βγ}=S

^{γβ}

_{α};

A

^{α}B

_{β}=A

^{α}B

_{β}g

^{α}

_{β}

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.