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This beginners' course provides students with a general and sufficiently easy to grasp theory of the Kurzweil-Henstock integral. The integral is indeed more general than Lebesgue's in RN, but its construction is rather simple, since it makes use of Riemann sums which, being geometrically viewable, are more easy to be understood. The theory is developed also for functions of several variables, and for differential forms, as well, finally leading to the celebrated Stokes-Cartan formula. In the appendices, differential calculus in RN is reviewed, with the theory of differentiable manifolds. Also, the Banach-Tarski paradox is presented here, with a complete proof, a rather peculiar argument for this type of monographs.
"The author maintains here the exposition at a very didactic level, trying to avoid as much as possible unnecessary technicalities, which is a big advantage of this book. ... In my point of view, the presented book is a useful tool for all mathematicians (not only for students) and I find it regrettable that this book was not written when I was a student." (Andrey Zahariev, zbMath 1410.26007, 2019)