Pullback of the metric from R3 to S2

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In summary, the pullback of the metric from R3 to S2 is the process of mapping a metric from three-dimensional Euclidean space onto a two-dimensional sphere. This is done using a coordinate transformation, allowing for the study of curved spaces with the tools of flat geometry. The pullback is important in fields such as differential geometry and physics and can be visualized through various methods. However, it has limitations, such as only being applicable to surfaces that can be embedded in three-dimensional space and potentially losing some information during the transformation process.
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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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