integration of 1-forms

Integration of 1-forms PhL 6.29.15 My reading of Buck has come to a crashing halt on page 410 with their claim that equation A is obvious. This equation says ∫γ ω* = ∫γ* ω ω = a 1-form After struggling for 2-3 days trying to understand this equation (which Buck's say is so obvious that they don't even discuss the equation other than to refer to an earlier section which has nothing to do with this equation), I need some other source on this subject that is readable by me. I did find 153 pages Sjammar from Cornell. He has a whole chapter on integrating 1-forms, but he does not involve any kind of ω* object. His chapter on integration of 1-forms requires a knowledge of "pulling back forms" which is a subject addressed in his Chapter 3. But this subject is way deep into the subject of "exterior algebra" which I have not really studied except for the Buck stuff. Exterior Algebra is in fact a bona fide subject. Some buzzwords from wiki are functoriality multivectors manifolds exterior product = wedge product Hermann Grassman in 1844 came up with this stuff This stuff is all far off from my "integration problem". I have just found a 10-page paper by Tao which claims to be about differential forms and integration. It is short enough that perhaps I will read it right now to try to gain some incite. Update 7.11.15. I have read up to page 50 in Sjamaar and I have learned a lot and at this point I take a look at the Buck mysterious equation (7-31) quoted above. My first question now is: what does * mean?