The Kurzweil-Henstock integral and its differentials: A unified theory of integration on \(\mathbb R\) and \({\mathbb R}^n\) (Q2718896)

This book deals with the theory of Kurzweil-Henstock integration, but it is very different from other books on the same topic. The author gives traditional differential notation a solid foundation and develops a theory of differentials based on the Kurzweil-Henstock integration process. Most of the known results on integration are reformulated in terms of differentials. The relation between derivatives and differentials is also discussed. There is also an introductory chapter on the higher-dimensional case.NEWLINENEWLINENEWLINEAn interval-point function on an interval \(I\) is called a summant. For examples, for \([u,v]\subset I\) and \(t= u\) or \(t= v\), define NEWLINE\[NEWLINES([u, v],t)= f(t) (g(v)- g(u)),\quad T([u, v], t)= g(v)- g(u).NEWLINE\]NEWLINE Then \(S\) and \(T\) are summants. In this book, the associated point \(t\) is always one of the endpoints of \([u,v]\). Two summants \(S\), \(T\) are differential equivalent if the Kurzweil-Henstock integral of \(|S-T|\) is zero. A differential is an equivalence class under differential equivalence. For examples, if \(g(t)\) is the integral of \(f\) over \([a,t]\), then \(dg\) and \(G\) are differential equivalent, where \(dg([u,v],t)= g(v)- g(u)\) and \(G([u,v], t)= f(t)[v- u]\).