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Mathematics 3

par: L. AMEUR
Disponible
Pages 204
Dimensions 16*23,5
Publication 2025-05-06
Éditeur —
Papier : 1 000.00 DA Numérique: 500.00 DA

WRITING Mathematics 03 was born out of a deep observation during my teaching journey: many students approach this subject with a sense of dread. This particular module has gained a reputation for being challenging, and it often becomes an overwhelming source of anxiety for engineering students. Witnessing this fear first-hand motivated me to simplify the material and present it in a way that makes it accessible and engaging. This book is primarily intended for second-year undergraduate students in science and technology, but it is also valuable for students in mathematics, technical sciences, and computer science, especially within the L.M.D. system. Anyone with an interest in mastering calculation techniques will find this work relevant. The content assumes a basic understanding equivalent to that of first-year courses in science and technology, or mathematics and computer science. Throughout the writing process, my main challenge and guiding principle was to simplify complex mathematical concepts without losing depth. This approach mirrors my teaching philosophy at the university helping students overcome their fear by making mathematics more approachable and enjoyable. I would like to express my heartfelt thanks to God, whose guidance, inspiration, and grace gave me the strength, patience, and determination to complete this book. I am deeply grateful to my parents, my sister and my wife, who encouraged me constantly throughout this endeavor. Lastly, I extend my gratitude to my colleagues, whose thoughtful feedback was invaluable in helping me refine and complete this work.

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Foreword
v
List of Figures
vii
List of Tables
xi
General Introduction
ix
1 Simple and Multiple Integrals
1
Riemann Integral
1.1 Partition of an Interval
1.2 Upper and Lower Sums
1.3 Riemann Sums
1.4 Approximation of a Definite Integral
1.5 Riemann Integrability
1.6 Necessary and Sufficient Conditions
1.7 Properties of Riemann Integrals
1.8 Primitive of a Continuous Function
1.9 Integration Techniques
1.10 Applications
Double Integrals
2.1 Introduction
2.2 Double Integral over a Rectangle
2.3 Double Integral over a Non-rectangular Domain
2.4 Double Integrals using Change of Variables
2.5 Applications of Double Integrals
Triple Integrals
3.1 Introduction
3.2 Triple Integral on a Parallelepiped
3.3 Triple Integral over an Arbitrary Domain
3.4 Triple Integrals using Change of Variables
3.5 Applications of Triple Integrals
Exercises with Solutions
4
2 Improper Integrals
49
Convergence
1.1 Absolute Convergence
Improper Integrals of Functions with Constant Sign
2.1 Criterion of Majorized Convergence
2.2 Comparison Criterion
2.3 Criterion of Dominated Convergence
Reference Integrals
3
Exercises with Solutions
4
3 Differential Equations
65
Motivation for Studying Differential Equations
Classification of Differential Equations
2.1 Order of Differential Equations
2.2 Linearity of Differential Equations
2.3 Type of Functions Involved
First Order Differential Equations
3.1 Equations with Separate Variables
3.2 Separable Equation
3.3 Homogeneous Equations
3.4 Equations Reducible to Homogeneous Equations
3.5 Total Differential Equations
3.6 First Order Linear Differential Equation
3.7 Bernoulli Differential Equation
3.8 Riccati Differential Equation
Second Order Linear Differential Equations
4.1 Second Order Homogeneous Linear Equations with Constant Coefficients
4.2 Second Order Inhomogeneous Linear Equations with Constant Coefficients
Exercises with Solutions
4 Numerical Series and Series of Functions
Numerical Series
1.1 Definitions and Properties
1.2 Necessary Condition of Convergence
1.3 Operations on Numerical Series
1.4 Convergence of Series with Positive Terms
1.5 Alternating Series
1.6 Series with Terms of Arbitrary Signs
Series of Functions
2.1 Criteria for Pointwise Convergence
2.2 Absolute Convergence
2.3 Normal Convergence
2.4 Uniform Convergence
2.5 Relationship between Types of Convergence
2.6 Uniform Abel Criterion
Power Series
3.1 Calculation of the Radius of Convergence
3.2 Radius of Convergence of the Derivative and Primitive Series
Exercises with Solutions
5 Fourier Transform
1 Definitions and Properties
1.1 Determination of Fourier Coefficients
2 Fourier Series of Periodic Functions with Period not equal to 2π
2.1 Parseval Equality
3 Fourier Series in Complex Form
4 Dirichlet Conditions
5 Exercises with Solutions
6 Laplace Transform
1 The Direct Laplace Transformation
1.1 Existence of L
1.2 Definition and Properties
1.3 Some Elementary Functions
2 The Inverse Laplace Transformation
3 Laplace Transform of Derivatives
4 Application to the Resolution of Differential Equations
5 Exercises with Solutions
Bibliography
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