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This undergraduate textbook is based on lectures given by the author on the differential and integral calculus of functions of several real variables. The book has a modern approach and includes topics such as:
-The p-norms on vector space and their equivalence
-The Weierstrass and Stone-Weierstrass approximation theorems
-The differential as a linear functional; Jacobians, Hessians, and Taylor's theorem in several variables
-The Implicit Function Theorem for a system of equations, proved via Banach's Fixed Point Theorem
-Applications to Ordinary Differential Equations
-Line integrals and an introduction to surface integrals
This book features numerous examples, detailed proofs, as well as exercises at the end of sections. Many of the exercises have detailed solutions, making the book suitable for self-study.
Several Real Variables will be useful for undergraduate students in mathematics who have completed first courses in linear algebra and analysis of one real variable.
List of contents
Continuity.- Derivation.- Implicit Functions.- Integration.- Solutions.
About the author
The author is an expert in Spectral Theory and semigroups of operators, researching in this area for 50 years, publishing numerous papers and three monographs.
Summary
This undergraduate textbook is based on lectures given by the author on the differential and integral calculus of functions of several real variables. The book has a modern approach and includes topics such as:
•The p-norms on vector space and their equivalence
•The Weierstrass and Stone-Weierstrass approximation theorems
•The differential as a linear functional; Jacobians, Hessians, and Taylor's theorem in several variables
•The Implicit Function Theorem for a system of equations, proved via Banach’s Fixed Point Theorem
•Applications to Ordinary Differential Equations
•Line integrals and an introduction to surface integrals
This book features numerous examples, detailed proofs, as well as exercises at the end of sections. Many of the exercises have detailed solutions, making the book suitable for self-study.
Several Real Variables will be useful for undergraduate students in mathematics who have completed first courses in linear algebra and analysis of one real variable.
Report
"This textbook is excellent for self-study as it contains a lot of well-chosen examples and complete proofs. Each topic is followed by a set of suitable exercises, accompanied with detailed solutions, which complement the presented material significantly. ... The reader can obtain a very deep insight into analysis and related theories as the book provides deep understanding of principles." (Roza Aceska, zbMATH 1405.26003, 2019)
