Calculus to Analysis
Description
This book addresses the analysis of functions of a real variable and transitions from the standard calculus sequence to mathematical analysis. The author presents the limits and convergence of sequences of functions, illustrates the limitations of the Riemann integral, and discusses the need for a new integral: the Lebesgue integral. The fundamental concepts of the theory of calculus of one variable is presented in addition to limits, continuity, derivatives and its applications, and integrals and their applications. The tone and language of the book is kept as informal as possible along with the descriptions and examples to aid learning. The book is concise and presents single variable advanced calculus leading up to Fourier analysis. In addition, the book sets up sufficient background for a course in measure theory and Lebesgue integration.
Reviews / Votes
"The main originality of the book is that the development of the topics presented is made through activities proposed to the reader. The author gives definitions, statements in the form of short propositions, indications for proofs for each activity as well as several examples and some bibliographical indications. The result is a stimulating challenge for the reader and a positive contribution to his mathematical reasoning ability." (Julià Cufí, zbMATH 1573.26001, 2026)
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Person
Arturo Portnoy, Ph.D., is a Professor of Mathematical Sciences at the University of Puerto Rico at Mayagüez. He is also Co-Director of the Puerto Rico Mathematical Olympiads. He received his B.Sc. in Applied Mathematics at the Instituto Tecnológico Autónomo de México (1992) and his M.Sc. (1994) and Ph.D. (1997) at Rensselaer Polytechnic institute. His areas of interest include inverse problems, mathematical modeling, optimization, regularization, math education, and math Olympiads. He also explores the use of technology to increase efficiency, transparency, and consistency in teaching mathematics.
Content
Preface.- The Fundamentals.- Sequences, Limits, and Series.- Continuous Functions.- Differentiable Functions.- The Riemann Integral.- Sequences of Functions and their Convergence.- Fourier Series I.- Fourier Series II.- Conclusions.
