Stokes’ theorem. (English) Zbl 1059.26008

Macdonald, Alan

Real Anal. Exch. 27(2001-2002), No. 2, 739-747 (2002).

In a nice and elementary way the Stokes theorem on the cube \([0,1]^n\) is proved using the Henstock-Kurzweil integral based on regular (cube) partitions of \([0,1]^n\) which is known to integrate every derivative in the case of \(n=1\). The main advantage of the generalized gauge integral is demonstrated in the paper without going into technical details of complicated nature. The idea is close to the heuristic approach to the Stokes theorem used by physicists and the author’s approach is fully satisfactory from the rigorous viewpoint of mathematics in the simple case of the cube \([0,1]^n\). The paper is closed by showing that the integral definition of the differential of a given form (with differentiable coefficients) coincides with its “derivative” definition.

Reviewer: Štefan Schwabik (Praha)

MSC:

26B20	Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
28A99	Classical measure theory
58C35	Integration on manifolds; measures on manifolds
26A39	Denjoy and Perron integrals, other special integrals

Keywords:

Stokes theorem; Henstock-Kurzweil integral; gauge integral