I'm a student of Physics and I've been studying manifolds and calculus on such objects for a time. Usually when we deal with vector calculus there are books that bring one operational point of view. For example: the book Mathematical Methods for Physicists by George Arfken.
This book brings interpretations of all the objects, like the vectors themselves, the integrals, the operations and so on and in the same time shows how one operates with them in practice. How to manipulate those objects and even carry down computations with them.
Calculus on manifolds, on the other hand, is being a little more complicated. The reason is that all books I've found until now focus just on theorems and their proofs. There's nothing wrong with it, of course this is interesting as well, but what I really need is that operational view with interpretations and so on.
For what I've seem until now, when doing calculus on manifolds one gets one incredible amount of work just to do some computations: work out charts, prove they are bijections and homeomorphism, prove they are $C^k$ related and so on. This also confuses me, because it seems much more complicated than vector calculus even to get started, while many people say it's not.
So, where can I learn this operational point of view of calculus on manifolds? Meaning, learn to interpret objects like exterior derivatives, differential forms and their integrals, and in the same time learn how to in practice carry out operations with those objects?
Thanks very much in advance.