Overview
- Contains more than 570 exercises of varying difficulty
- Provides proofs of basic results on existence and regularity of solutions of ordinary differential equations
- Includes a full treatment of the inverse function theorem in several variables
- Emphasizes the importance of estimates and computation in analysis
Part of the book series: Springer Undergraduate Mathematics Series (SUMS)
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Table of contents(9 chapters)
Keywords
- use of euler maclaurin formula
- Euler-Maclaurin formula
- metric space theory
- metric space in real analysis
- fourier series
- power series
- infinite series and products
- smooth and analytic functions
- uniform approximation
- existence theorem for ODEs
- derivative of vector-valued map
- implicit function theorem
- smooth functions theory
- real analysis techniques
- real analysis books
- 26-01, 40-01, 26Axx, 26Bxx, 26B05, 26B10, 26Exx, 26E05, 26E10
- 33Bxx, 33B15, 34A12, 40Axx, 42Axx, 42A10, 54Exx, 54-01, 54E35
About this book
This book provides a rigorous introduction to the techniques and results of real analysis, metric spaces and multivariate differentiation, suitable for undergraduate courses.
Starting from the very foundations of analysis, it offers a complete first course in real analysis, including topics rarely found in such detail in an undergraduate textbook such as the construction of non-analytic smooth functions, applications of the Euler-Maclaurin formula to estimates, and fractal geometry. Drawing on the author’s extensive teaching and research experience, the exposition is guided by carefully chosen examples and counter-examples, with the emphasis placed on the key ideas underlying the theory. Much of the content is informed by its applicability: Fourier analysis is developed to the point where it can be rigorously applied to partial differential equations or computation, and the theory of metric spaces includes applications to ordinary differential equations andfractals.
Essential Real Analysis will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis. Numerous exercises of varying difficulty, including some suitable for group work or class discussion, make this book suitable for self-study as well as lecture courses.
Reviews
“This book contains a reasonably complete exposition of real analysis which is needed for beginning undergraduate-level students. … This is a well-written textbook with an abundance of worked examples and exercises that are intended for a first course in analysis. This book offers a sound grounding in analysis. In particular, it gives a solid base in real analysis from which progress to more advanced topics may be made.” (Teodora-Liliana Rădulescu, zbMATH 1379.26001, 2018)
Authors and Affiliations
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Engineering Mathematics Department, Merchant Venturers School of Engineering, Bristol University, United Kingdom
Michael Field
About the author
Bibliographic Information
Book Title: Essential Real Analysis
Authors: Michael Field
Series Title: Springer Undergraduate Mathematics Series
DOI: https://doi.org/10.1007/978-3-319-67546-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Softcover ISBN: 978-3-319-67545-9Published: 15 November 2017
eBook ISBN: 978-3-319-67546-6Published: 06 November 2017
Series ISSN: 1615-2085
Series E-ISSN: 2197-4144
Edition Number: 1
Number of Pages: XVII, 450
Number of Illustrations: 29 b/w illustrations, 1 illustrations in colour
Topics: Real Functions, Sequences, Series, Summability, Topology, Fourier Analysis