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I can never derive the prolongation formulas correctly when I want to prove the Lie group symmetries of PDEs. (If I'm lucky I get the transformed tangent bundle coordinate right and botch the rest.) I've gone through a number of textbooks and such in the past, but I haven't found any clear, non-horrendous method for reproducing the prolongation formulas for a symmetry generator on the go. Anyone have any hints/content/advice?