Georgiev S. Multiple Integrals in Calculus...2025

Textbook in PDF format

This book presents an introduction to the theory of functions of several variables. The book is primarily intended for senior undergraduate students and beginning graduate students of engineering and science courses. Students in mathematical and physical sciences will find many sections of direct relevance.
In Chapter 1, the concepts of Jordan measure and measurable sets are introduced. Some of their properties are deduced. Multiple integrals are defined and explored. Some criteria for integrability are given. Some mean value theorems are proved. In Chapter 2, some methods for computing multiple integrals are provided. The Taylor formula is deduced. Linear maps on measurable sets are investigated. Metric properties of differentiable maps are deduced. In Chapter 3, improper multiple integrals are introduced. Some criteria for their convergence are deduced. Absolute convergence of improper multiple integrals is defined and investigated. In Chapter 4, we introduce curvilinear integrals of the first and second kinds. Some of their basic properties are deduced. The Green theorem is formulated and proved. In Chapter 5, surface integrals of the first and second kinds are defined, and some of their properties are listed. The Gauss–Ostrogradsky and Stokes formulas are formulated and proved. Solutions, hints, and answers to the exercises and problems are given