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im doing im im not going to do all the details but as i said in office hours the change of variables theorem is one of the big theorems in mathematics that everybody uses and its actually going to be crucial for us to understand how to do integrals of things on manifolds coming up soon but if you actually poll most people that have phds in mathematics very few of them actually have been through a proof of this there when you get to when you get to 8 000 level real analysis you learn a more powerful notion of integration called levate integration and using the vague integrals the proof of this theorem becomes easier but even so most graduate forces dont bother taking the time to explicitly do it so i want you to have some idea why the theorem works im not going to do all the details im not going to hold you responsible for any of the details other than the first thing were about to do which is something were going to use all over the place in chapters chapter 6. um so i remind