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.
Best Answer
I honestly don't think skipping vector calculus to be a particularly good idea. Yes, you can certainly do it and make do by jumping immediately to calculus on manifolds, but you're definitely going to have a severe gap in your understanding by having not taken the time to sit down and do computations that you might consider too simple, just because at some point you have to get your hands dirty with calculations and you need to do it early on so you understand what's going on later. Yes, abstraction is great and allows you to prove some truly striking things, but pretty much every professor I've had, in both undergraduate and graduate work, agrees that you don't understand the material if you can't do the computations regardless of how well you can prove something.
To the question itself: Spivak has a mixture of some straight computational problems and good theoretical problems, but the book being so short there are not nearly enough problems, in my opinion, to get a really solid grasp of vector calculus and calculus on (Euclidean) submanifolds. His goal is not to give you a solid calculus textbook or to even give a solid introduction to manifolds, but rather to make sense of Stoke's Theorem as quickly as possible and he addresses at least one topic (integration on chains) which as far as I'm aware is pretty outdated.
In terms of using a textbook to self-study vector calculus, I think you should read a combination of Lang's book, Munkre's Analysis on Manifolds (which is similar to Spivak though a little bit more drawn out and has a few more computational exercises), and honestly any decent multivariable standard calculus textbook like Hubbard or even Stewart to use just for basic problem solving. Both have a lot of problems to solve, and should give you a solid computational background in vector calculus that you'll need.