This book is designed to be an introductory course to some basic chapters of Advanced Mathematics for Engineering and Physics students, researchers in different branches of Applied Mathematics and anyone wanting to improve their mathematical knowledge by a clear, live, self-contained and motivated text.
Here, one can find different topics, such as differential (first order or higher order) equations, systems of differential equations, Fourier series, Fourier and Laplace transforms, partial differential equations, some basic facts and applications of the calculus of variations and, last but not least, an original and more intuitive introduction to probability theory. All these topics are carefully introduced, with complete proofs, motivations, examples, applications, problems and exercises, which are completely solved at the end of the book.We added a generous supplementary material (11.1) with a self-contained and complete introduction to normed, metric and Hilbert spaces. Since we used some topics from complex function theory, we also introduced in Chapter 11 a section (11.2) with the basic facts in this important field.
What a reader needs for a complete understanding of this book? For a deep understanding of this book, it is required to take a course in undergraduate calculus and linear algebra. We mostly tried to use the engineering intuition instead of insisting on mathematical tricks. The main feature of the material presented here is its clarity, motivation and the genuine desire of the authors to make extremely transparent the "mysterious" mathematical tools that are used to describe and organize the great variety of impressions that come to the searching mind, from the infinite complexity of Nature.The book is recommended not only to engineering and physics students or researchers but also to junior students in mathematics because it shows the connection between pure mathematics and physical phenomena, which always supply motivations for mathematical discoveries.
Author(s): Sever Angel Popescu, Marilena Jianu
Publisher: Springer
Year: 2023
Language: English
Pages: 832
City: Cham
Preface
Contents
Basic Notations
Sets
Hyperbolic Functions
Euler Integrals
1 First-Order Differential Equations
1.1 Introduction to Ordinary Differential Equations
1.2 Separable Equations
1.3 Homogeneous Equations
1.4 First-Order Linear Differential Equations
1.5 Bernoulli Equations
1.6 Riccati Equations
1.7 Exact Differential Equations
1.8 Lagrange Equations and Clairaut Equations
1.9 Existence and Uniqueness of Solution of the Cauchy Problem
1.10 Exercises
2 Higher-Order Differential Equations
2.1 Introduction
2.2 Homogeneous Linear Differential Equations of Order n
2.3 Non-Homogeneous Linear Differential Equations of Order n
2.4 Homogeneous Linear Equations with Constant Coefficients
2.5 Nonhomogeneous Linear Equations with Constant Coefficients
2.6 Euler Equations
2.7 Exercises
3 Systems of Differential Equations
3.1 Introduction
3.2 First-Order Systems and Differential Equations of Order n
3.3 Linear Systems of Differential Equations
3.4 Linear Systems with Constant Coefficients
3.4.1 The Homogeneous Case (the Algebraic Method)
3.4.2 The Non-Homogeneous Case (the Method of Undetermined Coefficients)
3.4.2.1 The Diagonalizable Case
3.4.2.2 The Non-Diagonalizable Case
3.4.3 Matrix Exponential and Linear Systems with Constant Coefficients
3.4.3.1 Fundamental Matrix
3.4.3.2 Matrix Exponential
3.4.3.3 The Exponential of a Diagonalizable Matrix
3.4.3.4 The Exponential of a Nondiagonalizable Matrix
3.4.4 Elimination Method for Linear Systems with Constant Coefficients
3.5 Autonomous Systems of Differential Equations
3.6 First-Order Partial Differential Equations
3.6.1 Linear Homogeneous First-Order PDE
3.6.2 Quasilinear First-Order Partial Differential Equations
3.7 Exercises
4 Fourier Series
4.1 Introduction: Periodic, Piecewise Smooth Functions
4.1.1 Periodic Functions
4.1.2 Piecewise Continuous and Piecewise Smooth Functions
4.2 Fourier Series Expansions
4.2.1 Series of Functions
4.2.2 A Basic Trigonometric System
4.2.3 Fourier Coefficients
4.3 Orthogonal Systems of Functions
4.3.1 Inner Product
4.3.2 Best Approximation in the Mean: Bessel's Inequality
4.4 The Convergence of Fourier Series
4.5 Differentiation and Integration of the Fourier Series
4.6 The Convergence in the Mean: Complete Systems
4.7 Examples of Fourier Expansions
4.8 The Complex form of the Fourier Series
4.9 Exercises
5 Fourier Transform
5.1 Improper Integrals
5.2 The Fourier Integral Formula
5.3 The Fourier Transform
5.4 Solving Linear Differential Equations
5.5 Moments Theorems
5.6 Sampling Theorem
5.7 Discrete Fourier Transform
5.8 Exercises
6 Laplace Transform
6.1 Introduction
6.2 Properties of the Laplace Transform
6.3 Inverse Laplace Transform
6.4 Solving Linear Differential Equations
6.5 The Dirac Delta Function
6.6 Exercises
7 Second-Order Partial Differential Equations
7.1 Classification: Canonical Form
7.2 The Wave Equation
7.2.1 Infinite Vibrating String: D'Alembert Formula
7.2.2 Finite Vibrating String: Fourier Method
7.2.3 Laplace Transform Method for the Vibrating String
7.2.4 Vibrations of a Rectangular Membrane: Two-Dimensional Wave Equation
7.3 Vibrations of a Simply Supported Beam: Fourier Method
7.4 The Heat Equation
7.4.1 Modeling the Heat Flow from a Body in Space
7.4.2 Heat Flow in a Finite Rod: Fourier Method
7.4.3 Heat Flow in an Infinite Rod
7.4.4 Heat Flow in a Rectangular Plate
7.5 The Laplace's Equation
7.5.1 Dirichlet Problem for a Rectangle
7.5.2 Dirichlet Problem for a Disk
7.6 Exercises
8 Introduction to the Calculus of Variations
8.1 Classical Variational Problems
8.2 General Frame of Calculus of Variations
8.3 The Case F[y]=abF(x,y,y) dx
8.4 The Case F[y]=ab F(x, y, y,…,y(n)) dx
8.5 The Case F[y1,…,yn]=abF(x,y1,…,yn,y1,…,yn) dx
8.6 The Case F[z]=@汥瑀瑯步渠D F (x,y,z,∂z∂x, ∂z∂y)dxdy
8.7 Isoperimetric Problems and Geodesic Problems
8.7.1 Isoperimetric Problems
8.7.2 Geodesic Problems
8.8 Exercises
9 Elements of Probability Theory
9.1 Sample Space: Event Space
9.2 Probability Space
9.3 Conditional Probability: Bayes Formula
9.4 Discrete Random Variables
9.4.1 Random Variables
9.4.2 Expected Value; Moments
9.4.3 Variance
9.4.4 Discrete Uniform Distribution
9.4.5 Bernoulli Distribution
9.4.6 Binomial Distribution
9.4.7 Poisson Distribution
9.4.8 Geometric Distribution
9.5 Continuous Random Variables
9.5.1 The Probability Density Function; The Distribution Function
9.5.2 Expected Value, Moments and Variance for Continuous Random Variables
9.5.3 Characteristic Function
9.5.4 The Uniform Distribution
9.5.5 The Exponential Distribution
9.5.6 The Normal Distribution
9.5.7 Gamma Distribution
9.5.8 Chi-Squared Distribution
9.5.9 Student t-Distribution
9.6 Limit Theorems
9.7 Exercises
10 Answers and Solutions to Exercises
10.1 Chapter 1
10.2 Chapter 2
10.3 Chapter 3
10.4 Chapter 4
10.5 Chapter 5
10.6 Chapter 6
10.7 Chapter 7
10.8 Chapter 8
10.9 Chapter 9
11 Supplementary Materials
11.1 Normed, Metric and Hilbert Spaces
11.1.1 Normed Vector Spaces
11.1.2 Sequences and Series of Functions
11.1.3 Metric Spaces. Some Density Theorems
11.1.4 The Fields Q, R and C
11.1.5 Hilbert Spaces
11.1.6 Continuous Functions and Step Functions
11.1.7 Orthonormal Systems in a Hilbert Space
11.2 Complex Function Theory
11.2.1 Differentiability of Complex Functions
11.2.2 Integration of Complex Functions
11.2.3 Power Series Representation
11.2.4 Residue Theorem and Applications
Bibliography
Index
