Essentials of mathematical methods in science and engineering / Ş. Selçuk Bayın.

Includes bibliographical references and index.

Preface xxiii Acknowledgments xxix 1 Functional Analysis 1 1.1 Concept of Function 1 1.2 Continuity and Limits 3 1.3 Partial Differentiation 6 1.4 Total Differential 8 1.5 Taylor Series 9 1.6 Maxima and Minima of Functions 13 1.7 Extrema of Functions with Conditions 17 1.8 Derivatives and Differentials of Composite Functions 21 1.9 Implicit Function Theorem 23 1.10 Inverse Functions 28 1.11 Integral Calculus and the Definite Integral 30 1.12 Riemann Integral 32 1.13 Improper Integrals 35 1.14 Cauchy Principal Value Integrals 38 1.15 Integrals Involving a Parameter 40 1.16 Limits of Integration Depending on a Parameter 44 1.17 Double Integrals 45 1.18 Properties of Double Integrals 47 1.19 Triple and Multiple Integrals 48 References 49 Problems 49 2 Vector Analysis 55 2.1 Vector Algebra: Geometric Method 55 2.1.1 Multiplication of Vectors 57 2.2 Vector Algebra: Coordinate Representation 60 2.3 Lines and Planes 65 2.4 Vector Differential Calculus 67 2.4.1 Scalar Fields and Vector Fields 67 2.4.2 Vector Differentiation 69 2.5 Gradient Operator 70 2.5.1 Meaning of the Gradient 71 2.5.2 Directional Derivative 72 2.6 Divergence and Curl Operators 73 2.6.1 Meaning of Divergence and the Divergence Theorem 75 2.7 Vector Integral Calculus in Two Dimensions 79 2.7.1 Arc Length and Line Integrals 79 2.7.2 Surface Area and Surface Integrals 83 2.7.3 An Alternate Way to Write Line Integrals 84 2.7.4 Green’s Theorem 86 2.7.5 Interpretations of Green’s Theorem 88 2.7.6 Extension to Multiply Connected Domains 89 2.8 Curl Operator and Stokes’s Theorem 92 2.8.1 On the Plane 92 2.8.2 In Space 96 2.8.3 Geometric Interpretation of Curl 99 2.9 Mixed Operations with the Del Operator 99 2.10 Potential Theory 102 2.10.1 Gravitational Field of a Star 105 2.10.2 Work Done by Gravitational Force 106 2.10.3 Path Independence and Exact Differentials 108 2.10.4 Gravity and Conservative Forces 109 2.10.5 Gravitational Potential 111 2.10.6 Gravitational Potential Energy of a System 113 2.10.7 Helmholtz Theorem 115 2.10.8 Applications of the Helmholtz Theorem 116 2.10.9 Examples from Physics 120 References 123 Problems 123 3 Generalized Coordinates and Tensors 133 3.1 Transformations between Cartesian Coordinates 134 3.1.1 Basis Vectors and Direction Cosines 134 3.1.2 Transformation Matrix and Orthogonality 136 3.1.3 Inverse Transformation Matrix 137 3.2 Cartesian Tensors 139 3.2.1 Algebraic Properties of Tensors 141 3.2.2 Kronecker Delta and the Permutation Symbol 145 3.3 Generalized Coordinates 148 3.3.1 Coordinate Curves and Surfaces 148 3.3.2 Why Upper and Lower Indices 152 3.4 General Tensors 153 3.4.1 Einstein Summation Convention 156 3.4.2 Line Element 157 3.4.3 Metric Tensor 157 3.4.4 How to Raise and Lower Indices 158 3.4.5 Metric Tensor and the Basis Vectors 160 3.4.6 Displacement Vector 161 3.4.7 Line Integrals 162 3.4.8 Area Element in Generalized Coordinates 164 3.4.9 Area of a Surface 165 3.4.10 Volume Element in Generalized Coordinates 169 3.4.11 Invariance and Covariance 171 3.5 Differential Operators in Generalized Coordinates 171 3.5.1 Gradient 171 3.5.2 Divergence 172 3.5.3 Curl 174 3.5.4 Laplacian 178 3.6 Orthogonal Generalized Coordinates 178 3.6.1 Cylindrical Coordinates 179 3.6.2 Spherical Coordinates 184 References 189 Problems 189 4 Determinants and Matrices 197 4.1 Basic Definitions 197 4.2 Operations with Matrices 198 4.3 Submatrix and Partitioned Matrices 204 4.4 Systems of Linear Equations 207 4.5 Gauss’s Method of Elimination 208 4.6 Determinants 211 4.7 Properties of Determinants 214 4.8 Cramer’s Rule 216 4.9 Inverse of a Matrix 221 4.10 Homogeneous Linear Equations 224 References 225 Problems 225 5 Linear Algebra 233 5.1 Fields and Vector Spaces 233 5.2 Linear Combinations, Generators, and Bases 236 5.3 Components 238 5.4 Linear Transformations 241 5.5 Matrix Representation of Transformations 242 5.6 Algebra of Transformations 244 5.7 Change of Basis 246 5.8 Invariants under Similarity Transformations 247 5.9 Eigenvalues and Eigenvectors 248 5.10 Moment of Inertia Tensor 257 5.11 Inner Product Spaces 262 5.12 The Inner Product 262 5.13 Orthogonality and Completeness 265 5.14 Gram–Schmidt Orthogonalization 267 5.15 Eigenvalue Problem for Real Symmetric Matrices 268 5.16 Presence of Degenerate Eigenvalues 270 5.17 Quadratic Forms 276 5.18 Hermitian Matrices 279 5.19 Matrix Representation of Hermitian Operators 283 5.20 Functions of Matrices 284 5.21 Function Space and Hilbert Space 286 5.22 Dirac’s Bra and Ket Vectors 287 References 288 Problems 289 6 Practical Linear Algebra 293 6.1 Systems of Linear Equations 294 6.1.1 Matrices and Elementary Row Operations 295 6.1.2 Gauss-Jordan Method 295 6.1.3 Information From the Row-Echelon Form 300 6.1.4 Elementary Matrices 301 6.1.5 Inverse by Gauss-Jordan Row-Reduction 302 6.1.6 Row Space, Column Space, and Null Space 303 6.1.7 Bases for Row, Column, and Null Spaces 307 6.1.8 Vector Spaces Spanned by a Set of Vectors 310 6.1.9 Rank and Nullity 312 6.1.10 Linear Transformations 315 6.2 Numerical Methods of Linear Algebra 317 6.2.1 Gauss-Jordan Row-Reduction and Partial Pivoting 317 6.2.2 LU-Factorization 321 6.2.3 Solutions of Linear Systems by Iteration 325 6.2.4 Interpolation 328 6.2.5 Power Method for Eigenvalues 331 6.2.6 Solution of Equations 333 6.2.7 Numerical Integration 343 References 349 Problems 350 7 Applications of Linear Algebra 355 7.1 Chemistry and Chemical Engineering 355 7.1.1 Independent Reactions and Stoichiometric Matrix 356 7.1.2 Independent Reactions from a Set of Species 359 7.2 Linear Programming 362 7.2.1 The Geometric Method 363 7.2.2 The Simplex Method 367 7.3 Leontief Input–Output Model of Economy 375 7.3.1 Leontief Closed Model 375 7.3.2 Leontief Open Model 378 7.4 Applications to Geometry 381 7.4.1 Orbit Calculations 382 7.5 Elimination Theory 383 7.5.1 Quadratic Equations and the Resultant 384 7.6 Coding Theory 388 7.6.1 Fields and Vector Spaces 388 7.6.2 Hamming (7,4) Code 390 7.6.3 Hamming Algorithm for Error Correction 393 7.7 Cryptography 396 7.7.1 Single-Key Cryptography 396 7.8 Graph Theory 399 7.8.1 Basic Definition 399 7.8.2 Terminology 400 7.8.3 Walks, Trails, Paths and Circuits 402 7.8.4 Trees and Fundamental Circuits 404 7.8.5 Graph Operations 404 7.8.6 Cut Sets and Fundamental Cut Sets 405 7.8.7 Vector Space Associated with a Graph 407 7.8.8 Rank and Nullity 409 7.8.9 Subspaces in WG 410 7.8.10 Dot Product and Orthogonal vectors 411 7.8.11 Matrix Representation of Graphs 413 7.8.12 Dominance Directed Graphs 417 7.8.13 Gray Codes in Coding Theory 419 References 419 Problems 420 8 Sequences and Series 425 8.1 Sequences 426 8.2 Infinite Series 430 8.3 Absolute and Conditional Convergence 431 8.3.1 Comparison Test 431 8.3.2 Limit Comparison Test 431 8.3.3 Integral Test 431 8.3.4 Ratio Test 432 8.3.5 Root Test 432 8.4 Operations with Series 436 8.5 Sequences and Series of Functions 438 8.6 M-Test for Uniform Convergence 441 8.7 Properties of Uniformly Convergent Series 441 8.8 Power Series 443 8.9 Taylor Series and Maclaurin Series 446 8.10 Indeterminate Forms and Series 447 References 448 Problems 448 9 Complex Numbers and Functions 453 9.1 The Algebra of Complex Numbers 454 9.2 Roots of a Complex Number 458 9.3 Infinity and the Extended Complex Plane 460 9.4 Complex Functions 463 9.5 Limits and Continuity 465 9.6 Differentiation in the Complex Plane 467 9.7 Analytic Functions 470 9.8 Harmonic Functions 471 9.9 Basic Differentiation Formulas 474 9.10 Elementary Functions 475 9.10.1 Polynomials 475 9.10.2 Exponential Function 476 9.10.3 Trigonometric Functions 477 9.10.4 Hyperbolic Functions 478 9.10.5 Logarithmic Function 479 9.10.6 Powers of Complex Numbers 481 9.10.7 Inverse Trigonometric Functions 483 References 483 Problems 484 10 Complex Analysis 491 10.1 Contour Integrals 492 10.2 Types of Contours 494 10.3 The Cauchy–Goursat Theorem 497 10.4 Indefinite Integrals 500 10.5 Simply and Multiply Connected Domains 502 10.6 The Cauchy Integral Formula 503 10.7 Derivatives of Analytic Functions 505 10.8 Complex Power Series 506 10.8.1 Taylor Series with the Remainder 506 10.8.2 Laurent Series with the Remainder 510 10.9 Convergence of Power Series 514 10.10 Classification of Singular Points 514 10.11 Residue Theorem 517 References 522 Problems 522 11 Ordinary Differential Equations 527 11.1 Basic Definitions for Ordinary Differential Equations 528 11.2 First-Order Differential Equations 530 11.2.1 Uniqueness of Solution 530 11.2.2 Methods of Solution 532 11.2.3 Dependent Variable is Missing 532 11.2.4 Independent Variable is Missing 532 11.2.5 The Case of Separable f(x, y) 532 11.2.6 Homogeneous f(x, y) of Zeroth Degree 533 11.2.7 Solution When f(x, y) is a Rational Function 533 11.2.8 Linear Equations of First-order 535 11.2.9 Exact Equations 537 11.2.10 Integrating Factors 539 11.2.11 Bernoulli Equation 542 11.2.12 Riccati Equation 543 11.2.13 Equations that Cannot Be Solved for y' 546 11.3 Second-Order Differential Equations 548 11.3.1 The General Case 549 11.3.2 Linear Homogeneous Equations with Constant Coefficients 551 11.3.3 Operator Approach 556 11.3.4 Linear Homogeneous Equations with Variable Coefficients 557 11.3.5 Cauchy–Euler Equation 560 11.3.6 Exact Equations and Integrating Factors 561 11.3.7 Linear Nonhomogeneous Equations 564 11.3.8 Variation of Parameters 564 11.3.9 Method of Undetermined Coefficients 566 11.4 Linear Differential Equations of Higher Order 569 11.4.1 With Constant Coefficients 569 11.4.2 With Variable Coefficients 570 11.4.3 Nonhomogeneous Equations 570 11.5 Initial Value Problem and Uniqueness of the Solution 571 11.6 Series Solutions: Frobenius Method 571 11.6.1 Frobenius Method and First-order Equations 581 References 582 Problems 582 12 Second-Order Differential Equations and Special Functions 589 12.1 Legendre Equation 590 12.1.1 Series Solution 590 12.1.2 Effect of Boundary Conditions 593 12.1.3 Legendre Polynomials 594 12.1.4 Rodriguez Formula 596 12.1.5 Generating Function 597 12.1.6 Special Values 599 12.1.7 Recursion Relations 600 12.1.8 Orthogonality 601 12.1.9 Legendre Series 603 12.2 Hermite Equation 606 12.2.1 Series Solution 606 12.2.2 Hermite Polynomials 610 12.2.3 Contour Integral Representation 611 12.2.4 Rodriguez Formula 612 12.2.5 Generating Function 613 12.2.6 Special Values 614 12.2.7 Recursion Relations 614 12.2.8 Orthogonality 616 12.2.9 Series Expansions in Hermite Polynomials 618 12.3 Laguerre Equation 619 12.3.1 Series Solution 620 12.3.2 Laguerre Polynomials 621 12.3.3 Contour Integral Representation 622 12.3.4 Rodriguez Formula 623 12.3.5 Generating Function 623 12.3.6 Special Values and Recursion Relations 624 12.3.7 Orthogonality 624 12.3.8 Series Expansions in Laguerre Polynomials 625 References 626 Problems 626 13 Bessel’s Equation and Bessel Functions 629 13.1 Bessel’s Equation and Its Series Solution 630 13.1.1 Bessel Functions J±m(x), Nm(x), and H(1,2)m (x) 634 13.1.2 Recursion Relations 639 13.1.3 Generating Function 639 13.1.4 Integral Definitions 641 13.1.5 Linear Independence of Bessel Functions 642 13.1.6 Modified Bessel Functions Im(x) and Km(x) 644 13.1.7 Spherical Bessel Functions jl(x), nl(x), and h(1,2)l (x) 645 13.2 Orthogonality and the Roots of Bessel Functions 648 13.2.1 Expansion Theorem 652 13.2.2 Boundary Conditions for the Bessel Functions 652 References 656 Problems 656 14 Partial Differential Equations and Separation of Variables 661 14.1 Separation of Variables in Cartesian Coordinates 662 14.1.1 Wave Equation 665 14.1.2 Laplace Equation 666 14.1.3 Diffusion and Heat Flow Equations 671 14.2 Separation of Variables in Spherical Coordinates 673 14.2.1 Laplace Equation 677 14.2.2 Boundary Conditions for a Spherical Boundary 678 14.2.3 Helmholtz Equation 682 14.2.4 Wave Equation 683 14.2.5 Diffusion and Heat Flow Equations 684 14.2.6 Time-Independent Schrödinger Equation 685 14.2.7 Time-Dependent Schrödinger Equation 685 14.3 Separation of Variables in Cylindrical Coordinates 686 14.3.1 Laplace Equation 688 14.3.2 Helmholtz Equation 689 14.3.3 Wave Equation 690 14.3.4 Diffusion and Heat Flow Equations 691 References 701 Problems 701 15 Fourier Series 705 15.1 Orthogonal Systems of Functions 705 15.2 Fourier Series 711 15.3 Exponential Form of the Fourier Series 712 15.4 Convergence of Fourier Series 713 15.5 Sufficient Conditions for Convergence 715 15.6 The Fundamental Theorem 716 15.7 Uniqueness of Fourier Series 717 15.8 Examples of Fourier Series 717 15.8.1 Square Wave 717 15.8.2 Triangular Wave 719 15.8.3 Periodic Extension 720 15.9 Fourier Sine and Cosine Series 721 15.10 Change of Interval 722 15.11 Integration and Differentiation of Fourier Series 723 References 724 Problems 724 16 Fourier and Laplace Transforms 727 16.1 Types of Signals 727 16.2 Spectral Analysis and Fourier Transforms 730 16.3 Correlation with Cosines and Sines 731 16.4 Correlation Functions and Fourier Transforms 735 16.5 Inverse Fourier Transform 736 16.6 Frequency Spectrums 736 16.7 Dirac-Delta Function 738 16.8 A Case with Two Cosines 739 16.9 General Fourier Transforms and Their Properties 740 16.10 Basic Definition of Laplace Transform 743 16.11 Differential Equations and Laplace Transforms 746 16.12 Transfer Functions and Signal Processors 748 16.13 Connection of Signal Processors 750 References 753 Problems 753 17 Calculus of Variations 757 17.1 A Simple Case 758 17.2 Variational Analysis 759 17.2.1 Case I: The Desired Function is Prescribed at the End Points 761 17.2.2 Case II: Natural Boundary Conditions 762 17.3 Alternate Form of Euler Equation 763 17.4 Variational Notation 765 17.5 A More General Case 767 17.6 Hamilton’s Principle 772 17.7 Lagrange’s Equations of Motion 773 17.8 Definition of Lagrangian 777 17.9 Presence of Constraints in Dynamical Systems 779 17.10 Conservation Laws 783 References 784 Problems 784 18 Probability Theory and Distributions 789 18.1 Introduction to Probability Theory 790 18.1.1 Fundamental Concepts 790 18.1.2 Basic Axioms of Probability 791 18.1.3 Basic Theorems of Probability 791 18.1.4 Statistical Definition of Probability 794 18.1.5 Conditional Probability and Multiplication Theorem 795 18.1.6 Bayes’ Theorem 796 18.1.7 Geometric Probability and Buffon’s Needle Problem 798 18.2 Permutations and Combinations 800 18.2.1 The Case of Distinguishable Balls with Replacement 800 18.2.2 The Case of Distinguishable Balls without Replacement 801 18.2.3 The Case of Indistinguishable Balls 802 18.2.4 Binomial and Multinomial Coefficients 803 18.3 Applications to Statistical Mechanics 804 18.3.1 Boltzmann Distribution for Solids 805 18.3.2 Boltzmann Distribution for Gases 807 18.3.3 Bose–Einstein Distribution for Perfect Gases 808 18.3.4 Fermi–Dirac Distribution 810 18.4 Statistical Mechanics and Thermodynamics 811 18.4.1 Probability and Entropy 811 18.4.2 Derivation of β 812 18.5 Random Variables and Distributions 814 18.6 Distribution Functions and Probability 817 18.7 Examples of Continuous Distributions 819 18.7.1 Uniform Distribution 819 18.7.2 Gaussian or Normal Distribution 820 18.7.3 Gamma Distribution 821 18.8 Discrete Probability Distributions 821 18.8.1 Uniform Distribution 822 18.8.2 Binomial Distribution 822 18.8.3 Poisson Distribution 824 18.9 Fundamental Theorem of Averages 825 18.10 Moments of Distribution Functions 826 18.10.1 Moments of the Gaussian Distribution 827 18.10.2 Moments of the Binomial Distribution 827 18.10.3 Moments of the Poisson Distribution 829 18.11 Chebyshev’s Theorem 831 18.12 Law of Large Numbers 832 References 833 Problems 834 19 Information Theory 841 19.1 Elements of Information Processing Mechanisms 844 19.2 Classical Information Theory 846 19.2.1 Prior Uncertainty and Entropy of Information 848 19.2.2 Joint and Conditional Entropies of Information 851 19.2.3 Decision Theory 854 19.2.4 Decision Theory and Game Theory 856 19.2.5 Traveler’s Dilemma and Nash Equilibrium 862 19.2.6 Classical Bit or Cbit 866 19.2.7 Operations on Cbits 869 19.3 Quantum Information Theory 871 19.3.1 Basic Quantum Theory 872 19.3.2 Single-Particle Systems and Quantum Information 878 19.3.3 Mach–Zehnder Interferometer 880 19.3.4 Mathematics of the Mach–Zehnder Interferometer 882 19.3.5 Quantum Bit or Qbit 886 19.3.6 The No-Cloning Theorem 889 19.3.7 Entanglement and Bell States 890 19.3.8 Quantum Dense Coding 895 19.3.9 Quantum Teleportation 896 References 900 Problems 901 Further Reading 907 Index 915