Pullback of the metric from R3 to S2

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The discussion focuses on the confusion surrounding the calculations presented in sections 5.12 and 5.13 of a specific document. The user questions the accuracy of the matrix of partials, suggesting it should be the transpose of the displayed version. Additionally, the user seeks clarification on representing the components of the matrix multiplication AB in relation to matrices A and B, indicating a need for a deeper understanding of matrix operations and summation indices.

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nigelscott
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I am looking at this document I do not understand how the author gets 5.12 and 5.13 on page 133. I think the matrix of partials should be the transpose of the one shown. Even so I still can't figure out how you get 5.13. Any help would be appreciated.
 
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Look at the summation indices in 5.13. How would you represent the components of the matrix multiplication ##AB## in terms of the components of the matrices ##A## and ##B##?
 

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