Advanced Calculus: Functions of Several Variables (Mathematical Monographs)

This book is a modification of the lectures Professor Roman Sikorski gave at Warsaw University in 1963/64 in place of the traditional course in the calculus of functions of several variables. More precisely, only Chapters IV-IX, which are the main part of the book, were the subject of the lectures. Chapters I-III, containing a necessary background from set theory, linear algebra and topology, are added for completeness, to avoid the necessity of referring to other books. Chapter X contains information, without proofs, about the further development or generalization of some material discussed in Chapters IV-IX. To understand this book it is enough to know the fundamental notions from the calculus of functions of one real variable, and the basic properties of determinants from algebra. A knowledge of analytic geometry is also assumed. All set-theoretical, algebraical and topological notions needed to understand the main part of the book are given in Chapters I-III. The reader familiar with the most fundamental notions from set, theory and topology can omit Chapters I and III. The reader familiar with linear algebra should look through Chapter II because of the terminology and notation. Many examples illustrating the main topics are given in exercises which are added at the end of almost every section. There are only a few exercises involving calculation of multiple integrals and finding extremal points of functions because traditional calculus texts contain enough problems of this type.