The Riemann Approach to Integration Local Geometric Theory [Paperback]

A detailed exposition of generalised RiemannStieltjes integrals.This book presents a detailed and mostly elementary exposition of the generalised Riemann-Stieltjes integrals discovered by Henstock, Kurzweil, and McShane more than thirty years ago. Aside from classical results, it contains some recent developments connected with lipeomorphic change of variable and the divergence theorem for discontinuously differentiable vector fields.This book presents a detailed and mostly elementary exposition of the generalised Riemann-Stieltjes integrals discovered by Henstock, Kurzweil, and McShane more than thirty years ago. Aside from classical results, it contains some recent developments connected with lipeomorphic change of variable and the divergence theorem for discontinuously differentiable vector fields.This book presents a detailed and mostly elementary exposition of the generalized Riemann-Stieltjes integrals discovered by Henstock, Kurzweil, and McShane. Along with the classical results, it contains some recent developments connected with lipeomorphic change of variables and the divergence theorem for discontinuously differentiable vector fields.Preface; Acknowledgments; Part I. One-Dimensional Integration: 1. Preliminaries; 2. The McShane integral; 3. Measure and measurability; 4. Integrable functions; 5. Descriptive definition; 6. The Henstock-Kurzweil integral; Part II. Multi-Dimensional Integration: 7. Preliminaries; 8. The McShane integral; 9. Descriptive definition; 10. Change of variables; 11. The gage integral; 12. The F-integral; 13. Recent developments; Bibliography; List of symbols; Index. The author presents a detailed exposition of some recent (Riemann!) approaches to generalized integrals of real-valued functions on compact intervals in Euclidean spaces Rm (m>=1)...The author presents this theory in a very complete, thorough manner. R.G. Bartle, Mathematical Reviews