The Riemann approach to integration : local geometric theory / Washek F. Pfeffer

bibliogr. ref. (p. [293]-295) and index

Traditionally, the divergence theorem or Gauss’ theorem was proved for continuously differentiable vector fields. Various extensions have been made by Mawhin, Pfeffer, Kurzweil, and Jarník. The present book gives a clear exposition of an account of the extension to discontinuously differentiable vector fields using the Henstock-Kurzweil integral.

The book is divided into two parts: one-dimensional and multidimensional. In each part, the McShane integral (an absolute version of the Henstock- Kurzweil integral) is studied in detail first. Nine versions of the divergence theorem are given in the book. In order to state the theorem in general form, the concepts of gages and calibers are introduced in Section 6.7 for the one-dimensional case in anticipation of its extension to the multidimensional one later. Roughly speaking, a gage is a positive function except for a thin set, e.g., a countable set; a caliber is a sequence of positive numbers which is used to control the size of the figures (elementary sets) in a partition. Then the gage integral is defined using Riemann sums which depend on a gage and a caliber. ... (Zentralblatt)