From Box 3.3, p. 85:(adsbygoogle = window.adsbygoogle || []).push({});

Since [tex]

S^{\alpha}_{\phantom{\alpha}\beta\gamma} = S(\omega^\alpha, e_\beta, e_\gamma)

[/tex]

and

sinceS[tex]=S^{\alpha}_{\phantom{\alpha}\beta\gamma}e_\alpha\otimes\omega^\beta\otimes\omega^\gamma[/tex]

is it then true that

S[tex]=S(\omega^\alpha, e_\beta, e_\gamma)e_\alpha\otimes\omega^\beta\otimes\omega^\gamma\ ?[/tex]

Also, to get a new tensor from an old tensor, one of the techniques is to contract two of the indexes with each other. Is this another form of contraction, namely:

[tex]T_\gamma = S^{\alpha}_{\phantom{\alpha}\alpha\gamma} = S^{\alpha}_{\phantom{\alpha}\beta\gamma}\eta^{\beta}_{\phantom{\beta}\alpha} = S^{\alpha}_{\phantom{\alpha}\beta\gamma}\eta^{\beta\lambda}\eta_{\lambda\alpha}\ ?[/tex]

Finally, why is the 1^{st}term on the rhs of this equation transposed??

[tex]\nabla([/tex]R[tex]\otimes[/tex]M[tex]) = (\nabla[/tex]R[tex]\otimes[/tex]M[tex])^T\ +\ [/tex]R[tex]\otimes\nabla[/tex]M

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# GRAVITAION, by MTW, Box 3.3

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