Advanced Calculus of a Single Variable

ISBN-13: 9783319278063 / Angielski / Twarda / 2016 / 382 str.

Tunc Geveci

Opis

The first semester of advanced calculus can be followed by a rigorous course in multivariable calculus and an introductory real analysis course that treats the Lebesgue integral and metric spaces, with special emphasis on Banach and Hilbert spaces.

Recenzje

"This is a textbook for a single variable advanced calculus course ... . This is a very traditional text on single variable advanced calculus, very readable. If I were teaching such a course this is a text to which I would give serious consideration." (G. A. Heuer, Mathematical Reviews, October, 2016)

"This volume is devoted to a thorough discussion of some basic concepts and theorems related to a beginning calculus course. ... The presentation is thorough and clear with many comments on the historical context of the problems and concepts. Requiring only basic knowledge of elementary calculus, this book presents the necessary material for students and professionals in various mathematics-related fields, such as engineering, statistics, and computer science, to explore real analysis." (Teodora-Liliana Radulescu, zbMATH 1339.26001, 2016)

Spis treści

Chapter 1: Real Numbers, Sequences and Limits.- Terminology and Notation.- Real Numbers.- The Limit of a Sequence.- The Cauchy Convergence Criterion.- The Least Upper Bound Principle.- Infinite Limits.- Chapter 2: Limits and Continuity of Functions.- Continuity.- The Limit of a Function at a Point.- Infinite Limits and Limits at Infinity.- The Intermediate Value Theorem.- Chapter 3: The Derivative.- The Derivative.- Local Linear Approximations and the Differential.- Rules of Differentiation.- The Mean Value Theorem.- L’Hôpital’s Rule.- Chapter 4: The Riemann Integral.- The Riemann Integral.- Basic Properties of the Integral.- The Fundamental Theorem of Calculus.- The Substitution Rule and Integration by Parts.- Improper Integrals: Part 1.- Improper Integrals: Part 2.- Chapter 5: Infinite Series.- Infinite Series of Numbers.- Convergence Tests for Infinite Series: Part 1.- Convergence Tests for Infinite Series: Part 2.- Chapter 6: Sequences and Series of Functions.- Sequences of Functions.- Infinite Series of Functions.- Power Series.- Taylor Series.- Another Look at Special Functions.

O autorze

Tunc Geveci is Professor Emeritus in the Department of Mathematics & Statistics at San Diego State University. His main publications have been in the fields of partial differential equations, numerical analysis, and the calculus of variations and optimal control.

Blurb

This advanced undergraduate textbook is based on a one-semester course on single variable calculus that the author has been teaching at San Diego State University for many years. The aim of this classroom-tested book is to deliver a rigorous discussion of the concepts and theorems that are dealt with informally in the first two semesters of a beginning calculus course. As such, students are expected to gain a deeper understanding of the fundamental concepts of calculus, such as limits (with an emphasis on ε-δ definitions), continuity (including an appreciation of the difference between mere pointwise and uniform continuity), the derivative (with rigorous proofs of various versions of L’Hôpital’s rule) and the Riemann integral (discussing improper integrals in-depth, including the comparison and Dirichlet tests). 

Success in this course is expected to prepare students for more advanced courses in real and complex analysis and this book will help to accomplish this. The first semester of advanced calculus can be followed by a rigorous course in multivariable calculus and an introductory real analysis course that treats the Lebesgue integral and metric spaces, with special emphasis on Banach and Hilbert spaces.