Differential Forms in Spivak vs Rudin

Question

Can anyone give me the gist of the difference of the treatment of Stokes' Theorem in Spivak versus baby Rudin (chapter 4 in spivak, chapter 10 in rudin)? I need to do some problems from Rudin but first I need to learn the material so I'm sort of pressed on time. I would prefer to read Spivak if it doesn't make a difference when doing the problems since he seems to use multilinear algebra which I think makes things clearer whereas Rudin's treatment seems a bit ad-hoc. Thanks.

Answer 1

I only went through Differential Forms recently reading through parts of Munkres, Spivak and Rudin and here are the differences that I noticed. Munkres and Spivak are pretty similar in the way they define everything however, Munkres does Stokes' Theorem in greater generality. Spivak and Rudin are radically different though. Whereas Spivak uses multilinear algebra, dual maps and the like to define what forms, pullbacks and exterior derivatives are, Rudin uses the "operational formula" approach to define everything which one can work with, but it's not very enlightening. Also, when defining integrals of forms and boundaries of regions Spivak uses cubes/cells whereas Rudin uses simplexes. In fact Rudin acknowledges these differences in his book and mentions:

Overall, I think Munkres' treatment is the most complete and user-friendly. Spivak's approach is similar but he is a bit terse (some people might prefer that though). Rudin should be avoided for Diff Forms.