Advanced Calculus of a Single Variable

Beschreibung

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 willhelp 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.

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 willhelp 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.

Carefully dissects key concepts such as limits of sequence, convergence & divergence of monotone sequences, infinite limits, derivatives, integrals, and series of real numbers Contextualizes subtle, commonly-misunderstood topics such as the notion of an infinite limit, the e-d definitions (for a better command of uniform versus pointwise continuity), error in local linear approximations, and integrability criteria Includes more than 120 exercises, with a solution manual available to instructors Includes supplementary material: sn.pub/extras

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Single Variable Advanced Calculus Riemann Integral Banach Spaces Hilbert Spaces Cauchy Sequence

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“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 Rădulescu, zbMATH 1339.26001, 2016)

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