Multiple Integrals in Calculus

The book consists of eight chapters, each focusing on different aspects of multiple integrals and related topics in mathematical analysis.In Chapter 1, multiple integrals are defined and developed. The Jordan measure in n-dimensional unit balls is introduced, along with the definition and criteria for multiple integrals, as well as their properties. Chapter 2 delves into advanced techniques for computing multiple integrals. It introduces the Taylor formula, discusses linear maps on measurable sets, and explores the metric properties of differentiable maps. In Chapter 3, we focus on improper multiple integrals and their properties. The chapter deduces criteria for the integrability of functions of several variables and develops concepts such as improper integrals of nonnegative functions, comparison criteria, and absolute convergence. Chapter 4 investigates the Stieltjes integral and its properties. Topics covered include the differentiation of monotone functions of finite variation and the Helly principle of choice, as well as continuous functions of finite variation. Chapter 5 addresses curvilinear integrals, defining line integrals of both the first and second kinds. It also discusses the independence of line integrals from the path of integration. In Chapter 6, surface integrals of the first and second kinds are introduced. The chapter presents the Gauss-Ostrogradsky theorem and Stokes’ formulas, along with advanced practical problems to practice these concepts.

Svetlin G. Georgiev works on various aspects of mathematics. His current research focuses on harmonic analysis, ordinary differential equations, partial differential equations, fractional calculus, time scale calculus, integral equations, numerical analysis, differential geometry, and dynamic geometry. Khaled Zennir: was born in Algeria 1982. He received his PhD in Mathematics in 2013 from Sidi Bel Abbès University, Algeria (Assist. professor). He is now associate Professor at Qassim University, KSA. His research interests lie in Nonlinear Hyperbolic Partial Differential Equations: Global Existence, Blow-Up, and Long Time Behavior.