Essentials of Mathematical Methods in Science and Engineering

Name: Essentials of Mathematical Methods in Science and Engineering
Brand: Wiley-Blackwell
Price: 141.99 EUR
Availability: OnlineOnly

Selcuk S. Bayin(Autor*in)

Wiley-Blackwell (Verlag)

2. Auflage

Erschienen am 27. November 2019

960 Seiten

E-Book

PDF mit Adobe-DRM

Systemvoraussetzungen

978-1-119-58023-2 (ISBN)

141,99 €inkl. 7% MwSt.

Systemvoraussetzungen

für PDF mit Adobe-DRM

E-Book Einzellizenz

Als Download verfügbar

Beschreibung

Weitere Details

Weitere Ausgaben

Inhalt

Cover
Title Page
Copyright
Contents
Preface
Acknowledgments
Chapter 1 Functional Analysis
1.1 Concept Of Function
1.2 Continuity and Limits
1.3 Partial Differentiation
1.4 Total Differential
1.5 Taylor Series
1.6 Maxima and Minima of Functions
1.7 Extrema of Functions with Conditions
1.8 Derivatives and Differentials of Composite Functions
1.9 Implicit Function Theorem
1.10 Inverse Functions
1.11 Integral Calculus and the Definite Integral
1.12 Riemann Integral
1.13 Improper Integrals
1.14 Cauchy Principal Value Integrals
1.15 Integrals Involving a Parameter
1.16 Limits of Integration Depending on a Parameter
1.17 Double Integrals
1.18 Properties of Double Integrals
1.19 Triple and Multiple Integrals
References
Chapter 2 Vector Analysis
2.1 Vector Algebra: Geometric Method
2.1.1 Multiplication of Vectors
2.2 Vector Algebra: Coordinate Representation
2.3 Lines and Planes
2.4 Vector Differential Calculus
2.4.1 Scalar Fields and Vector Fields
2.4.2 Vector Differentiation
2.5 Gradient Operator
2.5.1 Meaning of the Gradient
2.5.2 Directional Derivative
2.6 Divergence and Curl Operators
2.6.1 Meaning of Divergence and the Divergence Theorem
2.7 Vector Integral Calculus in Two Dimensions
2.7.1 Arc Length and Line Integrals
2.7.2 Surface Area and Surface Integrals
2.7.3 An Alternate Way to Write Line Integrals
2.7.4 Green's Theorem
2.7.5 Interpretations of Green's Theorem
2.7.6 Extension to Multiply Connected Domains
2.8 Curl Operator and Stokes's Theorem
2.8.1 On the Plane
2.8.2 In Space
2.8.3 Geometric Interpretation of Curl
2.9 Mixed Operations with the Del Operator
2.10 Potential Theory
2.10.1 Gravitational Field of a Star
2.10.2 Work Done by Gravitational Force
2.10.3 Path Independence and Exact Differentials
2.10.4 Gravity and Conservative Forces
2.10.5 Gravitational Potential
2.10.6 Gravitational Potential Energy of a System
2.10.7 Helmholtz Theorem
2.10.8 Applications of the Helmholtz Theorem
2.10.9 Examples from Physics
References
Chapter 3 Generalized Coordinates and Tensors
3.1 Transformations between Cartesian Coordinates
3.1.1 Basis Vectors and Direction Cosines
3.1.2 Transformation Matrix and Orthogonality
3.1.3 Inverse Transformation Matrix
3.2 Cartesian Tensors
3.2.1 Algebraic Properties of Tensors
3.2.2 Kronecker Delta and the Permutation Symbol
3.3 Generalized Coordinates
3.3.1 Coordinate Curves and Surfaces
3.3.2 Why Upper and Lower Indices
3.4 General Tensors
3.4.1 Einstein Summation Convention
3.4.2 Line Element
3.4.3 Metric Tensor
3.4.4 How to Raise and Lower Indices
3.4.5 Metric Tensor and the Basis Vectors
3.4.6 Displacement Vector
3.4.7 Line Integrals
3.4.8 Area Element in Generalized Coordinates
3.4.9 Area of a Surface
3.4.10 Volume Element in Generalized Coordinates
3.4.11 Invariance and Covariance
3.5 Differential Operators in Generalized Coordinates
3.5.1 Gradient
3.5.2 Divergence
3.5.3 Curl
3.5.4 Laplacian
3.6 Orthogonal Generalized Coordinates
3.6.1 Cylindrical Coordinates
3.6.2 Spherical Coordinates
References
Chapter 4 Determinants and Matrices
4.1 Basic Definitions
4.2 Operations with Matrices
4.3 Submatrix and Partitioned Matrices
4.4 Systems of Linear Equations
4.5 Gauss's Method of Elimination
4.6 Determinants
4.7 Properties of Determinants
4.8 Cramer's Rule
4.9 Inverse of a Matrix
4.10 Homogeneous Linear Equations
References
Chapter 5 Linear Algebra
5.1 Fields and Vector Spaces
5.2 Linear Combinations, Generators, and Bases
5.3 Components
5.4 Linear Transformations
5.5 Matrix Representation of Transformations
5.6 Algebra of Transformations
5.7 Change of Basis
5.8 Invariants under Similarity Transformations
5.9 Eigenvalues and Eigenvectors
5.10 Moment of Inertia Tensor
5.11 Inner Product Spaces
5.12 The Inner Product
5.13 Orthogonality and Completeness
5.14 Gram-Schmidt Orthogonalization
5.15 Eigenvalue Problem for Real Symmetric Matrices
5.16 Presence of Degenerate Eigenvalues
5.17 Quadratic Forms
5.18 Hermitian Matrices
5.19 Matrix Representation of Hermitian Operators
5.20 Functions of Matrices
5.21 Function Space and Hilbert Space
5.22 Dirac's Bra and Ket Vectors
References
Chapter 6 Practical Linear Algebra
6.1 Systems of Linear Equations
6.1.1 Matrices and Elementary Row Operations
6.1.2 Gauss-Jordan Method
6.1.3 Information From the Row-Echelon Form
6.1.4 Elementary Matrices
6.1.5 Inverse by Gauss-Jordan Row-Reduction
6.1.6 Row Space, Column Space, and Null Space
6.1.7 Bases for Row, Column, and Null Spaces
6.1.8 Vector Spaces Spanned by a Set of Vectors
6.1.9 Rank and Nullity
6.1.10 Linear Transformations
6.2 Numerical Methods of Linear Algebra
6.2.1 Gauss-Jordan Row-Reduction and Partial Pivoting
6.2.2 LU-Factorization
6.2.3 Solutions of Linear Systems by Iteration
6.2.4 Interpolation
6.2.5 Power Method for Eigenvalues
6.2.6 Solution of Equations
6.2.7 Numerical Integration
References
Chapter 7 Applications of Linear Algebra
7.1 Chemistry and Chemical Engineering
7.1.1 Independent Reactions and Stoichiometric Matrix
7.1.2 Independent Reactions from a Set of Species
7.2 Linear Programming
7.2.1 The Geometric Method
7.2.2 The Simplex Method
7.3 Leontief Input-Output Model of Economy
7.3.1 Leontief Closed Model
7.3.2 Leontief Open Model
7.4 Applications to Geometry
7.4.1 Orbit Calculations
7.5 Elimination Theory
7.5.1 Quadratic Equations and the Resultant
7.6 Coding Theory
7.6.1 Fields and Vector Spaces
7.6.2 Hamming (7,4) Code
7.6.3 Hamming Algorithm for Error Correction
7.7 Cryptography
7.7.1 Single-Key Cryptography
7.8 Graph Theory
7.8.1 Basic Definition
7.8.2 Terminology
7.8.3 Walks, Trails, Paths and Circuits
7.8.4 Trees and Fundamental Circuits
7.8.5 Graph Operations
7.8.6 Cut Sets and Fundamental Cut Sets
7.8.7 Vector Space Associated with a Graph
7.8.8 Rank and Nullity
7.8.9 Subspaces in WG
7.8.10 Dot Product and Orthogonal vectors
7.8.11 Matrix Representation of Graphs
7.8.12 Dominance Directed Graphs
7.8.13 Gray Codes in Coding Theory
References
Chapter 8 Sequences and Series
8.1 Sequences
8.2 Infinite Series
8.3 Absolute and Conditional Convergence
8.3.1 Comparison Test
8.3.2 Limit Comparison Test
8.3.3 Integral Test
8.3.4 Ratio Test
8.3.5 Root Test
8.4 Operations with Series
8.5 Sequences and Series of Functions
8.6 M-Test for Uniform Convergence
8.7 Properties of Uniformly Convergent Series
8.8 Power Series
8.9 Taylor Series and Maclaurin Series
8.10 Indeterminate Forms and Series
References
Chapter 9 Complex Numbers and Functions
9.1 The Algebra of Complex Numbers
9.2 Roots of a Complex Number
9.3 Infinity and the Extended Complex Plane
9.4 Complex Functions
9.5 Limits and Continuity
9.6 Differentiation in the Complex Plane
9.7 Analytic Functions
9.8 Harmonic Functions
9.9 Basic Differentiation Formulas
9.10 Elementary Functions
9.10.1 Polynomials
9.10.2 Exponential Function
9.10.3 Trigonometric Functions
9.10.4 Hyperbolic Functions
9.10.5 Logarithmic Function
9.10.6 Powers of Complex Numbers
9.10.7 Inverse Trigonometric Functions
References
Chapter 10 Complex Analysis
10.1 Contour Integrals
10.2 Types of Contours
10.3 The Cauchy-Goursat Theorem
10.4 Indefinite Integrals
10.5 Simply and Multiply Connected Domains
10.6 The Cauchy Integral Formula
10.7 Derivatives of Analytic Functions
10.8 Complex Power Series
10.8.1 Taylor Series with the Remainder
10.8.2 Laurent Series with the Remainder
10.9 Convergence of Power Series
10.10 Classification of Singular Points
10.11 Residue Theorem
References
Chapter 11 Ordinary Differential Equations
11.1 Basic Definitions for Ordinary Differential Equations
11.2 First-Order Differential Equations
11.2.1 Uniqueness of Solution
11.2.2 Methods of Solution
11.2.3 Dependent Variable is Missing
11.2.4 Independent Variable Is Missing
11.2.5 The Case of Separable f(x,y)
11.2.6 Homogeneous f(x,y) of Zeroth Degree
11.2.7 Solution when f(x,y) is a Rational Function
11.2.8 Linear Equations of First-order
11.2.9 Exact Equations
11.2.10 Integrating Factors
11.2.11 Bernoulli Equation
11.2.12 Riccati Equation
11.2.13 Equations that Cannot Be Solved for y´
11.3 Second-Order Differential Equations
11.3.1 The General Case
11.3.2 Linear Homogeneous Equations with Constant Coefficients
11.3.3 Operator Approach
11.3.4 Linear Homogeneous Equations with Variable Coefficients
11.3.5 Cauchy-Euler Equation
11.3.6 Exact Equations and Integrating Factors
11.3.7 Linear Nonhomogeneous Equations
11.3.8 Variation of Parameters
11.3.9 Method of Undetermined Coefficients
11.4 Linear Differential Equations of Higher Order
11.4.1 With Constant Coefficients
11.4.2 With Variable Coefficients
11.4.3 Nonhomogeneous Equations
11.5 Initial Value Problem and Uniqueness of the Solution
11.6 Series Solutions: Frobenius Method
11.6.1 Frobenius Method and First-order Equations
References
Chapter 12 Second-Order Differential Equations and Special Functions
12.1 Legendre Equation
12.1.1 Series Solution
12.1.2 Effect of Boundary Conditions
12.1.3 Legendre Polynomials
12.1.4 Rodriguez Formula
12.1.5 Generating Function
12.1.6 Special Values
12.1.7 Recursion Relations
12.1.8 Orthogonality
12.1.9 Legendre Series
12.2 Hermite Equation
12.2.1 Series Solution
12.2.2 Hermite Polynomials
12.2.3 Contour Integral Representation
12.2.4 Rodriguez Formula
12.2.5 Generating Function
12.2.6 Special Values
12.2.7 Recursion Relations
12.2.8 Orthogonality
12.2.9 Series Expansions in Hermite Polynomials
12.3 Laguerre Equation
12.3.1 Series Solution
12.3.2 Laguerre Polynomials
12.3.3 Contour Integral Representation
12.3.4 Rodriguez Formula
12.3.5 Generating Function
12.3.6 Special Values and Recursion Relations
12.3.7 Orthogonality
12.3.8 Series Expansions in Laguerre Polynomials
References
Chapter 13 Bessel's Equation and Bessel Functions
13.1 Bessel's Equation and Its Series Solution
13.1.1 Bessel Functions J±m(x),Nm(x), and Hm(1,2)(x)
13.1.2 Recursion Relations
13.1.3 Generating Function
13.1.4 Integral Definitions
13.1.5 Linear Independence of Bessel Functions
13.1.6 Modified Bessel Functions Im(x) and Km(x)
13.1.7 Spherical Bessel Functions jl(x),nl(x), and hl(1,2)(x)
13.2 Orthogonality and the Roots of Bessel Functions
13.2.1 Expansion Theorem
13.2.2 Boundary Conditions for the Bessel Functions
References
Chapter 14 Partial Differential Equations and Separation of Variables
14.1 Separation of Variables in Cartesian Coordinates
14.1.1 Wave Equation
14.1.2 Laplace Equation
14.1.3 Diffusion and Heat Flow Equations
14.2 Separation of Variables in Spherical Coordinates
14.2.1 Laplace Equation
14.2.2 Boundary Conditions for a Spherical Boundary
14.2.3 Helmholtz Equation
14.2.4 Wave Equation
14.2.5 Diffusion and Heat Flow Equations
14.2.6 Time-Independent Schrödinger Equation
14.2.7 Time-Dependent Schrödinger Equation
14.3 Separation of Variables in Cylindrical Coordinates
14.3.1 Laplace Equation
14.3.2 Helmholtz Equation
14.3.3 Wave Equation
14.3.4 Diffusion and Heat Flow Equations
References
Chapter 15 Fourier Series
15.1 Orthogonal Systems of Functions
15.2 Fourier Series
15.3 Exponential Form of the Fourier Series
15.4 Convergence of Fourier Series
15.5 Sufficient Conditions for Convergence
15.6 The Fundamental Theorem
15.7 Uniqueness of Fourier Series
15.8 Examples of Fourier Series
15.8.1 Square Wave
15.8.2 Triangular Wave
15.8.3 Periodic Extension
15.9 Fourier Sine and Cosine Series
15.10 Change of Interval
15.11 Integration and Differentiation of Fourier Series
References
Chapter 16 Fourier and Laplace Transforms
16.1 Types of Signals
16.2 Spectral Analysis and Fourier Transforms
16.3 Correlation with Cosines and Sines
16.4 Correlation Functions and Fourier Transforms
16.5 Inverse Fourier Transform
16.6 Frequency Spectrums
16.7 Dirac-Delta Function
16.8 A Case with Two Cosines
16.9 General Fourier Transforms and Their Properties
16.10 Basic Definition of Laplace Transform
16.11 Differential Equations and Laplace Transforms
16.12 Transfer Functions and Signal Processors
16.13 Connection of Signal Processors
References
Chapter 17 Calculus of Variations
17.1 A Simple Case
17.2 Variational Analysis
17.2.1 Case I: The Desired Function is Prescribed at the End Points
17.2.2 Case II: Natural Boundary Conditions
17.3 Alternate Form of Euler Equation
17.4 Variational Notation
17.5 A More General Case
17.6 Hamilton's Principle
17.7 Lagrange's Equations of Motion
17.8 Definition of Lagrangian
17.9 Presence of Constraints in Dynamical Systems
17.10 Conservation Laws
References
Chapter 18 Probability Theory and Distributions
18.1 Introduction to Probability Theory
18.1.1 Fundamental Concepts
18.1.2 Basic Axioms of Probability
18.1.3 Basic Theorems of Probability
18.1.4 Statistical Definition of Probability
18.1.5 Conditional Probability and Multiplication Theorem
18.1.6 Bayes' Theorem
18.1.7 Geometric Probability and Buffon's Needle Problem
18.2 Permutations and Combinations
18.2.1 The Case of Distinguishable Balls with Replacement
18.2.2 The Case of Distinguishable Balls Without Replacement
18.2.3 The Case of Indistinguishable Balls
18.2.4 Binomial and Multinomial Coefficients
18.3 Applications to Statistical Mechanics
18.3.1 Boltzmann Distribution for Solids
18.3.2 Boltzmann Distribution for Gases
18.3.3 Bose-Einstein Distribution for Perfect Gases
18.3.4 Fermi-Dirac Distribution
18.4 Statistical Mechanics and Thermodynamics
18.4.1 Probability and Entropy
18.4.2 Derivation of ß
18.5 Random Variables and Distributions
18.6 Distribution Functions and Probability
18.7 Examples of Continuous Distributions
18.7.1 Uniform Distribution
18.7.2 Gaussian or Normal Distribution
18.7.3 Gamma Distribution
18.8 Discrete Probability Distributions
18.8.1 Uniform Distribution
18.8.2 Binomial Distribution
18.8.3 Poisson Distribution
18.9 Fundamental Theorem of Averages
18.10 Moments of Distribution Functions
18.10.1 Moments of the Gaussian Distribution
18.10.2 Moments of the Binomial Distribution
18.10.3 Moments of the Poisson Distribution
18.11 Chebyshev's Theorem
18.12 Law of Large Numbers
References
Chapter 19 Information Theory
19.1 Elements of Information Processing Mechanisms
19.2 Classical Information Theory
19.2.1 Prior Uncertainty and Entropy of Information
19.2.2 Joint and Conditional Entropies of Information
19.2.3 Decision Theory
19.2.4 Decision Theory and Game Theory
19.2.5 Traveler's Dilemma and Nash Equilibrium
19.2.6 Classical Bit or Cbit
19.2.7 Operations on Cbits
19.3 Quantum Information Theory
19.3.1 Basic Quantum Theory
19.3.2 Single-Particle Systems and Quantum Information
19.3.3 Mach-Zehnder Interferometer
19.3.4 Mathematics of the Mach-Zehnder Interferometer
19.3.5 Quantum Bit or Qbit
19.3.6 The No-Cloning Theorem
19.3.7 Entanglement and Bell States
19.3.8 Quantum Dense Coding
19.3.9 Quantum Teleportation
References
Further Reading
Index
EULA

Systemvoraussetzungen

Als PDF speichern Als Link merken