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I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This question is about whether or not there is a more "foundational" subject under smooth manifolds as well. Is there a book or online resource that discusses the logic and methodology used in the study of smooth manifolds?