I have completed Velleman's book, 'How to prove it'. I have also worked through Apostol Vol.1. I have messed about with many rigorous single variable calculus textbooks, e.g.,Apostol, Spivak, Courant, Lang, etc. I had started working through Lang's, 'Calculus of several variables' but put it up to do a book like Edwards, 'Advanced Calculus:A Differential Forms Approach.' I now see that book to be a waste of time, because it is extremely 'hand-wavy' I can't stand mathematics out of physics texts and I will most certainly not tolerate the same from an actual math book. Therefore I am back to my starting point, I have finished Linear Algebra via Lang and have been perusing Hubbard and Hubbard's book and Artin. I do not like that Hubbard and Hubbard is so wordy. I don't really have the patience to put up with extremely long winded explanations of trivial facts just to get to the meat.
I would really like to work through Spivak's, 'Calculus on Manifolds' the problem being that I need to know if I can do without a book like Lang's, 'Calculus of Several Variables'? My goal is to get to manifolds and skip 'Vector Calculus' but I do not want to shortchange myself on computation if Spivak's book would leave me in that state.
I need help to determine if it is worth the time to work through Lang's book or can I just skip it, I want to be able to apply forms, etc to physics although I am a math major.