Essentials of Mathematical Methods in Science and Engineering

Contents in Brief
Contents
Preface
Acknowledgments
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
	Problems
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
		Problems
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 Coord
		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
	Problems
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
	Problems
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
	Problems
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
	Problems
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 W_G
		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
	Problems
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
	Problems
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
	Problems
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
	Problems
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
	Problems
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
	Problems
13. Bessel’s Equation and Bessel Functions
	13.1 Bessel’s Equation and Its Series Solution
		13.1.1 Bessel Functions J_{±m} (x), N_m(x), and H_m^{(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 I_m(x) and K_m(x)
		13.1.7 Spherical Bessel Functions j_l(x), n_l(x), and h_l^{(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
	Problems
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 Schr¨odinger Equation
		14.2.7 Time-Dependent Schr¨odinger 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
	Problems
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
	Problems
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
	Problems
17. Calculus of Variations
	17.1 A Simple Case
	17.2 Variational Analysis
		17.2.1 Case I: The Desired Function is Prescribed at theEnd 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
	Problems
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 β
	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
	Problems
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
	Problems
Further Reading
	Mathematical Methods Textbooks:
	Mathematical Methods with Computers:
	Calculus/Advanced Calculus:
	Linear Algebra and its Applications:
	Complex Calculus:
	Differential Equations:
	Calculus of Variations:
	Fourier Series, Integral Transforms and Signal Processing:
	Series and Special Functions:
	Mathematical Tables:
	Classical Mechanics:
	Quantum Mechanics:
	Electromagnetic Theory:
	Probability Theory:
	Information Theory:
Index