Titre :	Maths1 : Analysis and Algebra1
Type de document :	texte imprimé
Auteurs :	Aissa Benseghir, Auteur
Année de publication :	2022
Importance :	1 vol (95 p.)
Format :	29 cm
Langues :	Français (fre)
Catégories :	Publications pédagogiques:Mathématiaue P/P
Mots-clés :	Analysis  Algebra1  Numbers  Real functions  real variable  Linear algebra
Index. décimale :	510-Mathématique
Résumé :	Preface This mathematics course goes beyond certain developments within the strict frame- work of the program usually covered in the first year of the undergraduate cycle of higher education. We wanted to make it a reference document that engineering stu- dents can use in the rest of their studies to deepen or review the notions of algebra or analysis used in the teaching of applied mathematics for the master’s degree. In this work, we have endeavored to give precise definitions and present rigorous rea- soning without, however, seeking exhaustiveness. Furthermore, as far as possible, we have sought to motivate the concepts introduced and to illustrate them with examples, remarks and warnings in order to make learning more dynamic. This manuscript constitutes the essential part of Analysis 1 and Algebra 1 which I gave first year LMD science of matter. We will enhance the course with applications and motivations from physics and chemistry. It covers the essential elements of set theory, applications and relationships, internal laws, an introduction to general algebra such as groups, rings, fields. It then addresses the real functions of a real variable, in particular the notion of limit, its properties, the notion of continuity and differentiability of functions and finally we study the usual functions, these functions appear naturally in solving simple problems, especially those dealing with real-world topics in physics. It also covers an introduction of vector spaces given in the last chapter. We then formalize the abstract and fundamental concept in linear algebra as well as that of linear
Note de contenu :	Contents Preface i 1 General algebra 2 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Definitions of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Equivalence relation and Order relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Internal composition laws . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Group, Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 Rings, Sub-ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 Body, Subbody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.9 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.9.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.9.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Numbers 24 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Rational numbers and irrational numbers . . . . . . . . . . . . . . . . . 25 2.2.1 Decimal representation of rational and irrational numbers . . . . 25 2.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Real numbers and thier properties . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.2 Solving inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Least upper bounds and greatest lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.1 Upper and lower bounds . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2 Least upper bounds, greatest lower bounds . . . . . . . . . . . . 33 2.5 Reasoning by recurrence. . . . . . . . . . . . . . . . . . . . . 35 2.6 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Real functions with a real variable 40 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Notions of function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.2 Function operations . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.1 Continuity at a point . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.2 Continuity over an interval . . . . . . . . . . . . . . . . . . . . . 50 3.4.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5 Differentiation of a functions . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.6 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6.2 Graph of convex function . . . . . . . . . . . . . . . . . . . . . . 58 3.7 Derivative of usual functions . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 Some elementary functions 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Reciprocal circular functions . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.1 Brief reminders of trigonometric functions . . . . . . . . . . . . 68 4.2.2 Arcsine function . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.3 Arccosine function . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.4 Arctangent function . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.1 Definitions and first properties . . . . . . . . . . . . . . . . . . . 72 4.3.2 Addition formulas for hyperbolic functions . . . . . . . . . . . . 75 4.3.3 Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . 75 4.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5 Linear algebra 84 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Vector spaces , vector subspaces . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Linear application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Côte titre :	PM/0011
En ligne :	http://dspace.univ-setif.dz:8888/jspui/bitstream/123456789/4911/1/Maths1%20Analy [...]

Maths1 : Analysis and Algebra1 [texte imprimé] / Aissa Benseghir, Auteur . - 2022 . - 1 vol (95 p.) ; 29 cm.
Langues : Français (fre)

Catégories :	Publications pédagogiques:Mathématiaue P/P
Mots-clés :	Analysis  Algebra1  Numbers  Real functions  real variable  Linear algebra
Index. décimale :	510-Mathématique
Résumé :	Preface This mathematics course goes beyond certain developments within the strict frame- work of the program usually covered in the first year of the undergraduate cycle of higher education. We wanted to make it a reference document that engineering stu- dents can use in the rest of their studies to deepen or review the notions of algebra or analysis used in the teaching of applied mathematics for the master’s degree. In this work, we have endeavored to give precise definitions and present rigorous rea- soning without, however, seeking exhaustiveness. Furthermore, as far as possible, we have sought to motivate the concepts introduced and to illustrate them with examples, remarks and warnings in order to make learning more dynamic. This manuscript constitutes the essential part of Analysis 1 and Algebra 1 which I gave first year LMD science of matter. We will enhance the course with applications and motivations from physics and chemistry. It covers the essential elements of set theory, applications and relationships, internal laws, an introduction to general algebra such as groups, rings, fields. It then addresses the real functions of a real variable, in particular the notion of limit, its properties, the notion of continuity and differentiability of functions and finally we study the usual functions, these functions appear naturally in solving simple problems, especially those dealing with real-world topics in physics. It also covers an introduction of vector spaces given in the last chapter. We then formalize the abstract and fundamental concept in linear algebra as well as that of linear
Note de contenu :	Contents Preface i 1 General algebra 2 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Definitions of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Equivalence relation and Order relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Internal composition laws . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Group, Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 Rings, Sub-ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 Body, Subbody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.9 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.9.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.9.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Numbers 24 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Rational numbers and irrational numbers . . . . . . . . . . . . . . . . . 25 2.2.1 Decimal representation of rational and irrational numbers . . . . 25 2.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Real numbers and thier properties . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.2 Solving inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Least upper bounds and greatest lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.1 Upper and lower bounds . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2 Least upper bounds, greatest lower bounds . . . . . . . . . . . . 33 2.5 Reasoning by recurrence. . . . . . . . . . . . . . . . . . . . . 35 2.6 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Real functions with a real variable 40 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Notions of function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.2 Function operations . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.1 Continuity at a point . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.2 Continuity over an interval . . . . . . . . . . . . . . . . . . . . . 50 3.4.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5 Differentiation of a functions . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.6 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6.2 Graph of convex function . . . . . . . . . . . . . . . . . . . . . . 58 3.7 Derivative of usual functions . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 Some elementary functions 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Reciprocal circular functions . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.1 Brief reminders of trigonometric functions . . . . . . . . . . . . 68 4.2.2 Arcsine function . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.3 Arccosine function . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.4 Arctangent function . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.1 Definitions and first properties . . . . . . . . . . . . . . . . . . . 72 4.3.2 Addition formulas for hyperbolic functions . . . . . . . . . . . . 75 4.3.3 Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . 75 4.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5 Linear algebra 84 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Vector spaces , vector subspaces . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Linear application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Côte titre :	PM/0011
En ligne :	http://dspace.univ-setif.dz:8888/jspui/bitstream/123456789/4911/1/Maths1%20Analy [...]