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Aldo G. S. Ventre Calculus and Linear Algebra Fundamentals and Applications Calculus and Linear Algebra Aldo G. S. Ventre Calculus and Linear Algebra Fundamentals and Applications Aldo G. S. Ventre Napoli, Italy ISBN 978-3-031-20548-4 ISBN 978-3-031-20549-1 (eBook) https://doi.org/10.1007/978-3-031-20549-1 1st edition: © Wolters Kluwer Italia s.r.l., Milan, Italy 2021 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Diletta Lorenzo Giuseppe Rebecca Preface Some university courses envisage that mathematics coexists with diverse disciplines, such as history, arts, economics, natural sciences, architecture and design. Mathe- matics should be related to these disciplines, and the student should not detect objec- tives and contents in mathematics that are decentralized from his or her interests and from his or her world. Actually, for centuries and even a couple of millennia, and not only in the Western world, mathematics has also directed culture, philosophical and scientific speculation, has lived in company of arts, philosophy, architecture and music, in one rich and fertile environment full of perspectives. In fact, mathematics is the place of durable goods, rationality, wisdom, and emotions. The interested reader is the student who desires clarity. However, presenting concepts with simplicity is a demanding task: in every context there is a threshold beyond which simplification alters the meaning. The book is designed for students enrolled in first level courses, for which the knowledge of mathematics is functional to the entire educational programme (e.g., chemical or biological sciences, material science, information technology, and various engineering disciplines). In addition, the text can be a valid support for students enrolled in courses oriented to social sciences (e.g., economy and finance, marketing, management). This book is the result of my experience gained in the years of teaching Math- ematics, Geometry, and Calculus at the University of Naples Federico II, Univer- sity of Salerno, University of Trento and University of Campania, as well as in the context of editing, assembling and publishing texts for university courses. The struc- ture of this book is compliant to what is usually utilized in undergraduate courses of a British/American system: the results, theorems, statements, and exercises are proposed by illustrating the unifying principles of Mathematics in various prac- tical contexts and applications. Therefore, on the one hand attention is focused on the construction of solid and robust fundamentals through the presentation of the theoretical bases and the demonstrations of the theorems; on the other hand, wide selections of examples, problems and exercises, both posed and solved, corroborate and finalize the theoretical framework. vii viii Preface Clarity and appropriate examples aid the formal setting in the general presentation. Some theorems, whose proof is trivial, are simply stated and commented on. Also, regarding some demonstrations where rigor contrasts with clarity, I propose a reasoning by analogy or an intuitive presentation. My main concern was to make the book understandable for young students. The order of the chapters can be altered during the delivery of the course. For example, it is possible to premise the differential calculus to the linear algebra by anticipating the chapters from 17 to 22 before chapter 11. The presentation is accompanied by a set of examples and exercises with increasing complexity. With the exclusion of the first three chapters in which the discussion rests on logical and intuitive bases, each chapter contains sections dedicated to exercises and problems both solved and proposed. Advances in mathematics, since ancient times, are presented in their historical framework as a consequence of the relations of thoughts and their development. Scientific achievements and struggles are no coincidence, but a need for constructing a coherent world. Each chapter mentions some names linked to a mathematical fact or to a society and its time. Along this line, the past struggles and the triumphs of those who have contributed most to the development of mathematics become cornerstones that highlight the evolution of mathematics and science in general. The idea is to place the student in a dynamic context that, together with the fundamental formative elements of the subject, makes him or her to assume the role of an active participant in the evolution of ideas rather than a passive observer of the results, thus enabling him or her to develop an analytical as well as critical mindset. Napoli, Italy Aldo G. S. Ventre Acknowledgements The completion of this work would not have been possible without the precious help of my wife Vanda. During the composition of the manuscript, when difficulties arose, she feels my state of doubtful anxiety that leads me to close and be absorbed, thinking of nothing but a resolving sentence. In those moments the matter is out of control, but she puts all the pieces together and transmits calm and trust. I must thank Vanda for her kind support and understanding. Contents 1 Language. Sets ................................................ 1 1.1 Language .............................................. 1 1.2 Sets ................................................... 2 References .................................................... 5 2 Numbers and Propositions ..................................... 7 2.1 The Natural Numbers .................................... 7 2.1.1 Counting Problems ............................. 9 2.2 Prime Numbers ......................................... 10 2.2.1 Codes and Decoding ............................ 12 2.3 Integer Numbers ........................................ 13 2.4 Rational Numbers ....................................... 13 2.4.1 Representations of Rational Numbers .............. 15 2.4.2 The Numeration ................................ 16 2.5 The Real Numbers ...................................... 17 2.5.1 Density ....................................... 18 2.5.2 Closure of a Set with Respect to an Operations ...... 19 2.6 Abbreviated Notations ................................... 19 2.6.1 There is at Least One ... ......................... 19 2.7 The Implication ......................................... 20 2.7.1 Implication and Logical Equivalence .............. 20 2.7.2 The Theorem .................................. 22 2.7.3 Tertium Non Datur .............................. 24 2.7.4 Proofs in Science ............................... 25 2.7.5 Visual Proofs .................................. 26 2.7.6 The Inverse Th√eorem ............................ 28 2.7.7 Irrationality of 2 ............................... 28 2.7.8 The Pythagorean School ......................... 30 2.7.9 Socrates and the Diagonal of the Square ........... 30 ix x Contents 2.8 The Inductive Method and the Induction Principle ........... 31 2.8.1 Necessary Condition. Sufficient Condition ......... 33 2.9 Intuition ............................................... 34 2.10 Mathematics and Culture ................................. 35 2.10.1 On Education .................................. 35 2.10.2 Individual Study and Work ....................... 36 References .................................................... 36 3 Relations ..................................................... 39 3.1 Introduction ............................................ 39 3.2 Cartesian Product of Sets. Relations ........................ 40 3.3 Binary Relations ........................................ 41 3.3.1 Orderings ..................................... 43 3.3.2 The Power Set ................................. 44 3.3.3 Total Order .................................... 44 3.4 Preferences ............................................. 45 3.4.1 Indifference .................................... 45 3.5 Equivalence Relations ................................... 45 3.5.1 Partitions of a Set ............................... 47 3.5.2 Remainder Classes .............................. 47 References .................................................... 48 4 Euclidean Geometry ........................................... 49 4.1 Introduction ............................................ 49 4.2 First Axioms ........................................... 50 4.3 The Axiomatic Method .................................. 51 4.3.1 Further Axioms of Euclidean Geometry ............ 51 4.4 The Refoundation of Geometry ........................... 52 4.5 Geometric Figures ....................................... 54 4.5.1 Convex and Concave Figures ..................... 54 4.5.2 Angles ........................................ 55 4.5.3 Relations Between Lines and Planes ............... 59 4.5.4 Relations Between Planes ........................ 62 4.5.5 Projections .................................... 63 4.5.6 The Angle of a Line and a Plane .................. 65 4.5.7 Dihedrals ...................................... 65 4.5.8 Perpendicular Planes ............................ 67 4.5.9 Symmetries .................................... 68 4.5.10 Similar Polygons ............................... 69 4.6 Thales’ Theorem ........................................ 70 References .................................................... 71 Contents xi 5 Functions ..................................................... 73 5.1 Introduction ............................................ 73 5.2 Equipotent Sets. Infinite Sets, Finite Sets ................... 77 5.3 Hotel Hilbert ........................................... 78 5.4 Composite Functions .................................... 79 5.5 Restriction and Extension of a Function .................... 80 Bibliography .................................................. 80 6 The Real Line ................................................. 81 6.1 Introduction ............................................ 81 6.2 The Coordinate System of the Axis ........................ 81 6.2.1 The Measure of a Segment ....................... 82 6.2.2 The Coordinate System of an Axis ................ 82 6.3 Equalities and Identities. Equivalent Equations .............. 84 6.3.1 Examples ...................................... 85 6.3.2 Forming an Equation from Given Information ...... 86 6.4 Order in R ............................................. 87 6.4.1 Evaluating an Inequality to Making a Decision ...... 88 6.5 Intervals, Neighborhoods, Absolute Value .................. 89 6.5.1 Exercises ...................................... 92 6.6 The Extended Set of Real Numbers ........................ 93 6.6.1 Examples ...................................... 94 6.7 Upper Bounds and Lower Bounds ......................... 95 6.8 Commensurability and Real Numbers ...................... 98 6.9 Separate Sets and Contiguous Sets ......................... 99 Bibliography .................................................. 100 7 Real-Valued Functions of a Real Variable. The Line .............. 101 7.1 The Cartesian Plane ..................................... 101 7.1.1 Quadrants ..................................... 102 7.1.2 Distance ....................................... 103 7.2 Real-Valued Functions of a Real Variable ................... 104 7.2.1 Extrema of a Real-Valued Function ............... 104 7.2.2 The Graph of a Real-Valued Function ............. 105 7.2.3 Graph and Curve ............................... 106 7.3 Lines in the Cartesian Plane .............................. 107 7.3.1 The Constant Function .......................... 107 7.3.2 The Identical Function .......................... 108 7.3.3 The Function f : x→ kx ......................... 108 7.3.4 The Function f : x → kx + n ..................... 111 7.3.5 The Linear Equation ............................ 112 7.3.6 The Parametric Equations of the Line .............. 113 7.4 Parallel Lines ........................................... 115 7.4.1 Parallel Lines Represented by Parametric Equations ..................................... 115