Georgiev S. Differential and Integral 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.
The book contains six chapters. In Chapter 1, the set Rn is defined. n-dimensional ball, sphere, and rectangular neighborhood of a point are defined. Sequences in Rn are defined by their properties. Open, closed and compact sets in Rn are investigated. The Heine-Borel theorem is formulated and proved. In Chapter 2, functions of several variables are introduced. Limits of functions of several variables are defined with their properties are deducted. Continuous functions of several variables are investigated. Uniform continuity is introduced and explored. In Chapter 3, partial derivatives of higher order and differentials for functions of several variables and their properties are introduced. Criteria for the differentiability of functions of several variables are deducted. Gradient of a function explored. Directional derivatives are defined and investigated. In Chapter 4, higher order partial derivatives of functions of several variables are investigated. The minimum and maximum of functions of several variables are introduced and investigated. Implicit functions are defined and explored. The method of Lagrange multipliers is introduced. In Chapters 5 and 6, many exercises are stated and solved in detail, and solutions, hints, and answers to the problems are given