Essentials of Mathematical Methods in Science and Engineering

SELÇUK Ş. BAYIN, PHD, is Professor in the Institute of Applied Mathematics at the Middle East Technical University in Ankara, Turkey. He has over thirty years of academic experience on the use of mathematical methods in physics courses, and his current research focuses on general relativity and cosmology. He is the author of the first and second editions of Mathematical Methods in Science and Engineering, and Essentials of Mathematical Methods in Science and Engineering, from Wiley.

• Preface xxiiiAcknowledgments xxix1 Functional Analysis 11.1 Concept of Function 11.2 Continuity and Limits 31.3 Partial Differentiation 61.4 Total Differential 81.5 Taylor Series 91.6 Maxima and Minima of Functions 131.7 Extrema of Functions with Conditions 171.8 Derivatives and Differentials of Composite Functions 211.9 Implicit Function Theorem 231.10 Inverse Functions 281.11 Integral Calculus and the Definite Integral 301.12 Riemann Integral 321.13 Improper Integrals 351.14 Cauchy Principal Value Integrals 381.15 Integrals Involving a Parameter 401.16 Limits of Integration Depending on a Parameter 441.17 Double Integrals 451.18 Properties of Double Integrals 471.19 Triple and Multiple Integrals 48References 49Problems 492 Vector Analysis 552.1 Vector Algebra: Geometric Method 552.1.1 Multiplication of Vectors 572.2 Vector Algebra: Coordinate Representation 602.3 Lines and Planes 652.4 Vector Differential Calculus 672.4.1 Scalar Fields and Vector Fields 672.4.2 Vector Differentiation 692.5 Gradient Operator 702.5.1 Meaning of the Gradient 712.5.2 Directional Derivative 722.6 Divergence and Curl Operators 732.6.1 Meaning of Divergence and the Divergence Theorem 752.7 Vector Integral Calculus in Two Dimensions 792.7.1 Arc Length and Line Integrals 792.7.2 Surface Area and Surface Integrals 832.7.3 An Alternate Way to Write Line Integrals 842.7.4 Green’s Theorem 862.7.5 Interpretations of Green’s Theorem 882.7.6 Extension to Multiply Connected Domains 892.8 Curl Operator and Stokes’s Theorem 922.8.1 On the Plane 922.8.2 In Space 962.8.3 Geometric Interpretation of Curl 992.9 Mixed Operations with the Del Operator 992.10 Potential Theory 1022.10.1 Gravitational Field of a Star 1052.10.2 Work Done by Gravitational Force 1062.10.3 Path Independence and Exact Differentials 1082.10.4 Gravity and Conservative Forces 1092.10.5 Gravitational Potential 1112.10.6 Gravitational Potential Energy of a System 1132.10.7 Helmholtz Theorem 1152.10.8 Applications of the Helmholtz Theorem 1162.10.9 Examples from Physics 120References 123Problems 1233 Generalized Coordinates and Tensors 1333.1 Transformations between Cartesian Coordinates 1343.1.1 Basis Vectors and Direction Cosines 1343.1.2 Transformation Matrix and Orthogonality 1363.1.3 Inverse Transformation Matrix 1373.2 Cartesian Tensors 1393.2.1 Algebraic Properties of Tensors 1413.2.2 Kronecker Delta and the Permutation Symbol 1453.3 Generalized Coordinates 1483.3.1 Coordinate Curves and Surfaces 1483.3.2 Why Upper and Lower Indices 1523.4 General Tensors 1533.4.1 Einstein Summation Convention 1563.4.2 Line Element 1573.4.3 Metric Tensor 1573.4.4 How to Raise and Lower Indices 1583.4.5 Metric Tensor and the Basis Vectors 1603.4.6 Displacement Vector 1613.4.7 Line Integrals 1623.4.8 Area Element in Generalized Coordinates 1643.4.9 Area of a Surface 1653.4.10 Volume Element in Generalized Coordinates 1693.4.11 Invariance and Covariance 1713.5 Differential Operators in Generalized Coordinates 1713.5.1 Gradient 1713.5.2 Divergence 1723.5.3 Curl 1743.5.4 Laplacian 1783.6 Orthogonal Generalized Coordinates 1783.6.1 Cylindrical Coordinates 1793.6.2 Spherical Coordinates 184References 189Problems 1894 Determinants and Matrices 1974.1 Basic Definitions 1974.2 Operations with Matrices 1984.3 Submatrix and Partitioned Matrices 2044.4 Systems of Linear Equations 2074.5 Gauss’s Method of Elimination 2084.6 Determinants 2114.7 Properties of Determinants 2144.8 Cramer’s Rule 2164.9 Inverse of a Matrix 2214.10 Homogeneous Linear Equations 224References 225Problems 2255 Linear Algebra 2335.1 Fields and Vector Spaces 2335.2 Linear Combinations, Generators, and Bases 2365.3 Components 2385.4 Linear Transformations 2415.5 Matrix Representation of Transformations 2425.6 Algebra of Transformations 2445.7 Change of Basis 2465.8 Invariants under Similarity Transformations 2475.9 Eigenvalues and Eigenvectors 2485.10 Moment of Inertia Tensor 2575.11 Inner Product Spaces 2625.12 The Inner Product 2625.13 Orthogonality and Completeness 2655.14 Gram–Schmidt Orthogonalization 2675.15 Eigenvalue Problem for Real Symmetric Matrices 2685.16 Presence of Degenerate Eigenvalues 2705.17 Quadratic Forms 2765.18 Hermitian Matrices 2795.19 Matrix Representation of Hermitian Operators 2835.20 Functions of Matrices 2845.21 Function Space and Hilbert Space 2865.22 Dirac’s Bra and Ket Vectors 287References 288Problems 2896 Practical Linear Algebra 2936.1 Systems of Linear Equations 2946.1.1 Matrices and Elementary Row Operations 2956.1.2 Gauss-Jordan Method 2956.1.3 Information From the Row-Echelon Form 3006.1.4 Elementary Matrices 3016.1.5 Inverse by Gauss-Jordan Row-Reduction 3026.1.6 Row Space, Column Space, and Null Space 3036.1.7 Bases for Row, Column, and Null Spaces 3076.1.8 Vector Spaces Spanned by a Set of Vectors 3106.1.9 Rank and Nullity 3126.1.10 Linear Transformations 3156.2 Numerical Methods of Linear Algebra 3176.2.1 Gauss-Jordan Row-Reduction and Partial Pivoting 3176.2.2 LU-Factorization 3216.2.3 Solutions of Linear Systems by Iteration 3256.2.4 Interpolation 3286.2.5 Power Method for Eigenvalues 3316.2.6 Solution of Equations 3336.2.7 Numerical Integration 343References 349Problems 3507 Applications of Linear Algebra 3557.1 Chemistry and Chemical Engineering 3557.1.1 Independent Reactions and Stoichiometric Matrix 3567.1.2 Independent Reactions from a Set of Species 3597.2 Linear Programming 3627.2.1 The Geometric Method 3637.2.2 The Simplex Method 3677.3 Leontief Input–Output Model of Economy 3757.3.1 Leontief Closed Model 3757.3.2 Leontief Open Model 3787.4 Applications to Geometry 3817.4.1 Orbit Calculations 3827.5 Elimination Theory 3837.5.1 Quadratic Equations and the Resultant 3847.6 Coding Theory 3887.6.1 Fields and Vector Spaces 3887.6.2 Hamming (7,4) Code 3907.6.3 Hamming Algorithm for Error Correction 3937.7 Cryptography 3967.7.1 Single-Key Cryptography 3967.8 Graph Theory 3997.8.1 Basic Definition 3997.8.2 Terminology 4007.8.3 Walks, Trails, Paths and Circuits 4027.8.4 Trees and Fundamental Circuits 4047.8.5 Graph Operations 4047.8.6 Cut Sets and Fundamental Cut Sets 4057.8.7 Vector Space Associated with a Graph 4077.8.8 Rank and Nullity 4097.8.9 Subspaces in WG 4107.8.10 Dot Product and Orthogonal vectors 4117.8.11 Matrix Representation of Graphs 4137.8.12 Dominance Directed Graphs 4177.8.13 Gray Codes in Coding Theory 419References 419Problems 4208 Sequences and Series 4258.1 Sequences 4268.2 Infinite Series 4308.3 Absolute and Conditional Convergence 4318.3.1 Comparison Test 4318.3.2 Limit Comparison Test 4318.3.3 Integral Test 4318.3.4 Ratio Test 4328.3.5 Root Test 4328.4 Operations with Series 4368.5 Sequences and Series of Functions 4388.6 M-Test for Uniform Convergence 4418.7 Properties of Uniformly Convergent Series 4418.8 Power Series 4438.9 Taylor Series and Maclaurin Series 4468.10 Indeterminate Forms and Series 447References 448Problems 4489 Complex Numbers and Functions 4539.1 The Algebra of Complex Numbers 4549.2 Roots of a Complex Number 4589.3 Infinity and the Extended Complex Plane 4609.4 Complex Functions 4639.5 Limits and Continuity 4659.6 Differentiation in the Complex Plane 4679.7 Analytic Functions 4709.8 Harmonic Functions 4719.9 Basic Differentiation Formulas 4749.10 Elementary Functions 4759.10.1 Polynomials 4759.10.2 Exponential Function 4769.10.3 Trigonometric Functions 4779.10.4 Hyperbolic Functions 4789.10.5 Logarithmic Function 4799.10.6 Powers of Complex Numbers 4819.10.7 Inverse Trigonometric Functions 483References 483Problems 48410 Complex Analysis 49110.1 Contour Integrals 49210.2 Types of Contours 49410.3 The Cauchy–Goursat Theorem 49710.4 Indefinite Integrals 50010.5 Simply and Multiply Connected Domains 50210.6 The Cauchy Integral Formula 50310.7 Derivatives of Analytic Functions 50510.8 Complex Power Series 50610.8.1 Taylor Series with the Remainder 50610.8.2 Laurent Series with the Remainder 51010.9 Convergence of Power Series 51410.10 Classification of Singular Points 51410.11 Residue Theorem 517References 522Problems 52211 Ordinary Differential Equations 52711.1 Basic Definitions for Ordinary Differential Equations 52811.2 First-Order Differential Equations 53011.2.1 Uniqueness of Solution 53011.2.2 Methods of Solution 53211.2.3 Dependent Variable is Missing 53211.2.4 Independent Variable is Missing 53211.2.5 The Case of Separable f(x, y) 53211.2.6 Homogeneous f(x, y) of Zeroth Degree 53311.2.7 Solution When f(x, y) is a Rational Function 53311.2.8 Linear Equations of First-order 53511.2.9 Exact Equations 53711.2.10 Integrating Factors 53911.2.11 Bernoulli Equation 54211.2.12 Riccati Equation 54311.2.13 Equations that Cannot Be Solved for y' 54611.3 Second-Order Differential Equations 54811.3.1 The General Case 54911.3.2 Linear Homogeneous Equations with Constant Coefficients 55111.3.3 Operator Approach 55611.3.4 Linear Homogeneous Equations with Variable Coefficients 55711.3.5 Cauchy–Euler Equation 56011.3.6 Exact Equations and Integrating Factors 56111.3.7 Linear Nonhomogeneous Equations 56411.3.8 Variation of Parameters 56411.3.9 Method of Undetermined Coefficients 56611.4 Linear Differential Equations of Higher Order 56911.4.1 With Constant Coefficients 56911.4.2 With Variable Coefficients 57011.4.3 Nonhomogeneous Equations 57011.5 Initial Value Problem and Uniqueness of the Solution 57111.6 Series Solutions: Frobenius Method 57111.6.1 Frobenius Method and First-order Equations 581References 582Problems 58212 Second-Order Differential Equations and Special Functions 58912.1 Legendre Equation 59012.1.1 Series Solution 59012.1.2 Effect of Boundary Conditions 59312.1.3 Legendre Polynomials 59412.1.4 Rodriguez Formula 59612.1.5 Generating Function 59712.1.6 Special Values 59912.1.7 Recursion Relations 60012.1.8 Orthogonality 60112.1.9 Legendre Series 60312.2 Hermite Equation 60612.2.1 Series Solution 60612.2.2 Hermite Polynomials 61012.2.3 Contour Integral Representation 61112.2.4 Rodriguez Formula 61212.2.5 Generating Function 61312.2.6 Special Values 61412.2.7 Recursion Relations 61412.2.8 Orthogonality 61612.2.9 Series Expansions in Hermite Polynomials 61812.3 Laguerre Equation 61912.3.1 Series Solution 62012.3.2 Laguerre Polynomials 62112.3.3 Contour Integral Representation 62212.3.4 Rodriguez Formula 62312.3.5 Generating Function 62312.3.6 Special Values and Recursion Relations 62412.3.7 Orthogonality 62412.3.8 Series Expansions in Laguerre Polynomials 625References 626Problems 62613 Bessel’s Equation and Bessel Functions 62913.1 Bessel’s Equation and Its Series Solution 63013.1.1 Bessel Functions J±m(x), Nm(x), and H(1,2)m (x) 63413.1.2 Recursion Relations 63913.1.3 Generating Function 63913.1.4 Integral Definitions 64113.1.5 Linear Independence of Bessel Functions 64213.1.6 Modified Bessel Functions Im(x) and Km(x) 64413.1.7 Spherical Bessel Functions jl(x), nl(x), and h(1,2)l (x) 64513.2 Orthogonality and the Roots of Bessel Functions 64813.2.1 Expansion Theorem 65213.2.2 Boundary Conditions for the Bessel Functions 652References 656Problems 65614 Partial Differential Equations and Separation of Variables 66114.1 Separation of Variables in Cartesian Coordinates 66214.1.1 Wave Equation 66514.1.2 Laplace Equation 66614.1.3 Diffusion and Heat Flow Equations 67114.2 Separation of Variables in Spherical Coordinates 67314.2.1 Laplace Equation 67714.2.2 Boundary Conditions for a Spherical Boundary 67814.2.3 Helmholtz Equation 68214.2.4 Wave Equation 68314.2.5 Diffusion and Heat Flow Equations 68414.2.6 Time-Independent Schrödinger Equation 68514.2.7 Time-Dependent Schrödinger Equation 68514.3 Separation of Variables in Cylindrical Coordinates 68614.3.1 Laplace Equation 68814.3.2 Helmholtz Equation 68914.3.3 Wave Equation 69014.3.4 Diffusion and Heat Flow Equations 691References 701Problems 70115 Fourier Series 70515.1 Orthogonal Systems of Functions 70515.2 Fourier Series 71115.3 Exponential Form of the Fourier Series 71215.4 Convergence of Fourier Series 71315.5 Sufficient Conditions for Convergence 71515.6 The Fundamental Theorem 71615.7 Uniqueness of Fourier Series 71715.8 Examples of Fourier Series 71715.8.1 Square Wave 71715.8.2 Triangular Wave 71915.8.3 Periodic Extension 72015.9 Fourier Sine and Cosine Series 72115.10 Change of Interval 72215.11 Integration and Differentiation of Fourier Series 723References 724Problems 72416 Fourier and Laplace Transforms 72716.1 Types of Signals 72716.2 Spectral Analysis and Fourier Transforms 73016.3 Correlation with Cosines and Sines 73116.4 Correlation Functions and Fourier Transforms 73516.5 Inverse Fourier Transform 73616.6 Frequency Spectrums 73616.7 Dirac-Delta Function 73816.8 A Case with Two Cosines 73916.9 General Fourier Transforms and Their Properties 74016.10 Basic Definition of Laplace Transform 74316.11 Differential Equations and Laplace Transforms 74616.12 Transfer Functions and Signal Processors 74816.13 Connection of Signal Processors 750References 753Problems 75317 Calculus of Variations 75717.1 A Simple Case 75817.2 Variational Analysis 75917.2.1 Case I: The Desired Function is Prescribed at the End Points 76117.2.2 Case II: Natural Boundary Conditions 76217.3 Alternate Form of Euler Equation 76317.4 Variational Notation 76517.5 A More General Case 76717.6 Hamilton’s Principle 77217.7 Lagrange’s Equations of Motion 77317.8 Definition of Lagrangian 77717.9 Presence of Constraints in Dynamical Systems 77917.10 Conservation Laws 783References 784Problems 78418 Probability Theory and Distributions 78918.1 Introduction to Probability Theory 79018.1.1 Fundamental Concepts 79018.1.2 Basic Axioms of Probability 79118.1.3 Basic Theorems of Probability 79118.1.4 Statistical Definition of Probability 79418.1.5 Conditional Probability and Multiplication Theorem 79518.1.6 Bayes’ Theorem 79618.1.7 Geometric Probability and Buffon’s Needle Problem 79818.2 Permutations and Combinations 80018.2.1 The Case of Distinguishable Balls with Replacement 80018.2.2 The Case of Distinguishable Balls without Replacement 80118.2.3 The Case of Indistinguishable Balls 80218.2.4 Binomial and Multinomial Coefficients 80318.3 Applications to Statistical Mechanics 80418.3.1 Boltzmann Distribution for Solids 80518.3.2 Boltzmann Distribution for Gases 80718.3.3 Bose–Einstein Distribution for Perfect Gases 80818.3.4 Fermi–Dirac Distribution 81018.4 Statistical Mechanics and Thermodynamics 81118.4.1 Probability and Entropy 81118.4.2 Derivation of β 81218.5 Random Variables and Distributions 81418.6 Distribution Functions and Probability 81718.7 Examples of Continuous Distributions 81918.7.1 Uniform Distribution 81918.7.2 Gaussian or Normal Distribution 82018.7.3 Gamma Distribution 82118.8 Discrete Probability Distributions 82118.8.1 Uniform Distribution 82218.8.2 Binomial Distribution 82218.8.3 Poisson Distribution 82418.9 Fundamental Theorem of Averages 82518.10 Moments of Distribution Functions 82618.10.1 Moments of the Gaussian Distribution 82718.10.2 Moments of the Binomial Distribution 82718.10.3 Moments of the Poisson Distribution 82918.11 Chebyshev’s Theorem 83118.12 Law of Large Numbers 832References 833Problems 83419 Information Theory 84119.1 Elements of Information Processing Mechanisms 84419.2 Classical Information Theory 84619.2.1 Prior Uncertainty and Entropy of Information 84819.2.2 Joint and Conditional Entropies of Information 85119.2.3 Decision Theory 85419.2.4 Decision Theory and Game Theory 85619.2.5 Traveler’s Dilemma and Nash Equilibrium 86219.2.6 Classical Bit or Cbit 86619.2.7 Operations on Cbits 86919.3 Quantum Information Theory 87119.3.1 Basic Quantum Theory 87219.3.2 Single-Particle Systems and Quantum Information 87819.3.3 Mach–Zehnder Interferometer 88019.3.4 Mathematics of the Mach–Zehnder Interferometer 88219.3.5 Quantum Bit or Qbit 88619.3.6 The No-Cloning Theorem 88919.3.7 Entanglement and Bell States 89019.3.8 Quantum Dense Coding 89519.3.9 Quantum Teleportation 896References 900Problems 901Further Reading 907Index 915