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.
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##?