This book is devoted to a multi-dimensional version of a very classical problem: recovering a function from its derivative. An immediate application of our results yields the Gauss-Green and Stokes theorems of large generality.

The problem we consider has a long history. In dimension one, it was solved by Lebesgue for absolutely continuous functions, and by Denjoy and Perron (independently and by different means) for so called ACG* functions [75, Chapter 7, Section 8]. In higher dimensions, absolutely continuous functions become absolutely continuous measures, which can still be recovered from their Radon-Nikodym derivatives by means of the Lebesgue integral. A multi-dimensional analog of ACG* functions is more subtle, and has been defined only recently [18, 19].

There is no obvious extension of the Denjoy-Perron integral to higher dimensions. The early generalizations [4, 72] do not integrate partial derivatives of all differentiable functions, and give no indication how this can be achieved. Even the strikingly simple Riemannian definition of the Denjoy-Perron integral, obtained independently by Henstock [30] and Kurzweil [42], did not initially produce desirable results in higher dimensions [43, 52]. The first successful multi-dimensional generalization is due to Mawhin [50, 49], who modified the Henstock-Kurzweil definition so that the partial derivatives of each differentiable function are integrable and the Gauss-Green formula holds.