Studying Mathematics

The Beauty, the Toil and the Method

Book

1. Front Matter
Pages I-XVII
2. Part I

1. Front Matter
Pages 1-3
2. Marco Bramanti, Giancarlo Travaglini
Pages 5-12
3. Marco Bramanti, Giancarlo Travaglini
Pages 13-20
4. Marco Bramanti, Giancarlo Travaglini
Pages 21-35
5. Marco Bramanti, Giancarlo Travaglini
Pages 37-44
6. Marco Bramanti, Giancarlo Travaglini
Pages 45-59
7. Marco Bramanti, Giancarlo Travaglini
Pages 61-74
8. Marco Bramanti, Giancarlo Travaglini
Pages 75-90
9. Marco Bramanti, Giancarlo Travaglini
Pages 91-101
10. Marco Bramanti, Giancarlo Travaglini
Pages 103-133
3. Part II

1. Front Matter
Pages 135-138
2. Marco Bramanti, Giancarlo Travaglini
Pages 139-166
3. Marco Bramanti, Giancarlo Travaglini
Pages 167-200
4. Marco Bramanti, Giancarlo Travaglini
Pages 201-231
5. Marco Bramanti, Giancarlo Travaglini
Pages 233-264
6. Marco Bramanti, Giancarlo Travaglini
Pages 265-331
4. Part III

1. Front Matter
Pages 333-335
2. Marco Bramanti, Giancarlo Travaglini
Pages 337-354
3. Marco Bramanti, Giancarlo Travaglini
Pages 355-361

Introduction

This book is dedicated to preparing prospective college students for the study of mathematics.

It can be used at the end of high school or during the first year of college, for personal study or for introductory courses. It aims to set a meeting between two relatives who rarely speak to each other: the Mathematics of Beauty, which shows up in some popular books and films, and the Mathematics of Toil, which is widely known. Toil can be overcome through an appropriate method of work. Beauty will be found in the achievement of a way of thinking.

The first part concerns the mathematical language: the expressions “for all”, “there exists”, “implies”, “is false”, ...; what is a proof by contradiction; how to use indices, sums, induction.

The second part tackles specific difficulties: to study a definition, to understand an idea and apply it, to fix a slightly wrong argument, to discuss suggestions, to explain a proof.

The third part presents customary techniques and points of view in college mathematics.

The reader can choose one of three difficulty levels (A, B, C).

Keywords

mathematics education mathematics textbook university mathematics undergraduate mathematics mathematical exercises

Authors and affiliations

1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
2. 2.Dipartimento di Matematica e ApplicazioniUniversità di Milano-Bicocca, Edificio U5MilanoItaly

Marco Bramanti is Professor of Mathematics at the Politecnico di Milano. His current research interests include linear second order partial differential equations with nonnegative characteristic form, geometry of Hörmander's vector fields, real analysis, in particular singular integral theory. He has authored numerous research papers, one research monograph, and numerous textbooks.

Giancarlo Travaglini is Professor of Mathematics at the University of Milano-Bicocca. He is the author of numerous research papers and the author or the editor of several books in the areas of abstract harmonic analysis, Fourier analysis, discrepancy theory and mathematics education. His current mathematical interests include Fourier analysis, discrepancy theory, Radon transforms and didactics of mathematics.

Bibliographic information

• Book Title Studying Mathematics
• Book Subtitle The Beauty, the Toil and the Method
• Authors Marco Bramanti
Giancarlo Travaglini
• DOI https://doi.org/10.1007/978-3-319-91355-1
• Copyright Information Springer International Publishing AG, part of Springer Nature 2018
• Publisher Name Springer, Cham
• eBook Packages Education Education (R0)
• Hardcover ISBN 978-3-319-91354-4
• Softcover ISBN 978-3-030-08234-5
• eBook ISBN 978-3-319-91355-1
• Edition Number 1
• Number of Pages XVII, 398
• Number of Illustrations 129 b/w illustrations, 11 illustrations in colour
• Topics
• Buy this book on publisher's site