My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter 10 and the proofs and I can follow the logic and verify that they are true. However, I don't see why someone would come up with their definition and what makes them useful for building a theory of integration.
To rephrase this: What information exactly is encapsulated by the definition of differential forms and what makes them work out so nicely with respect to wedge products? Why is this the "right" formulation for an integration theory?