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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?

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    I answered a similar question on MathSE's 'sister site' mathoverflow a while back- you may be after something more in depth, but you may find the second and third paragraphs useful: mathoverflow.net/questions/25389/is-a-conceptual-explanation-possible-for-why-the-space-of-1-forms-on-a-manifold-c/25393#253932012-03-27
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    When integrating a function over a surface in R^3, a change of variables leads you to integrate a different function: the function multiplied by the abs. value of the Jacobian of the change of coordinates. This is even apparent in one-variable integration (u-substitution). So if you want to do integration in a coordinate-free way (say, on a manifold without natural coordinates), integration of functions is not well-defined. You want to integrate objects which change under a change of coordinates by mult. by the abs. value of the Jacobian. Those are differential forms (on oriented manifold).2012-03-28

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