{"product_id":"essentials-of-mathematical-methods-in-science-and-engineering-9781119580249","title":"Essentials of Mathematical Methods in Science and","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eA comprehensive introduction to the multidisciplinary applications of mathematical methods, revised and updated\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eThe second edition of \u003ci\u003eEssentials of Mathematical Methods in Science and Engineering\u003c\/i\u003e offers an introduction to the key mathematical concepts of advanced calculus, differential equations, complex analysis, and introductory mathematical physics for students in engineering and physics research. The book's approachable style is designed in a modular format with each chapter covering a subject thoroughly and thus can be read independently.\u003c\/p\u003e \u003cp\u003eThis updated second edition includes two new and extensive chapters that cover practical linear algebra and applications of linear algebra as well as a computer file that includes Matlab codes. To enhance understanding of the material presented, the text contains a collection of exercises at the end of each chapter. The author offers a coherent treatment of the topics with a style that makes the essential mathematic\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003c\/p\u003e\u003cp\u003ePreface xxiii\u003c\/p\u003e \u003cp\u003eAcknowledgments xxix\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Functional Analysis 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Concept of Function 1\u003c\/p\u003e \u003cp\u003e1.2 Continuity and Limits 3\u003c\/p\u003e \u003cp\u003e1.3 Partial Differentiation 6\u003c\/p\u003e \u003cp\u003e1.4 Total Differential 8\u003c\/p\u003e \u003cp\u003e1.5 Taylor Series 9\u003c\/p\u003e \u003cp\u003e1.6 Maxima and Minima of Functions 13\u003c\/p\u003e \u003cp\u003e1.7 Extrema of Functions with Conditions 17\u003c\/p\u003e \u003cp\u003e1.8 Derivatives and Differentials of Composite Functions 21\u003c\/p\u003e \u003cp\u003e1.9 Implicit Function Theorem 23\u003c\/p\u003e \u003cp\u003e1.10 Inverse Functions 28\u003c\/p\u003e \u003cp\u003e1.11 Integral Calculus and the Definite Integral 30\u003c\/p\u003e \u003cp\u003e1.12 Riemann Integral 32\u003c\/p\u003e \u003cp\u003e1.13 Improper Integrals 35\u003c\/p\u003e \u003cp\u003e1.14 Cauchy Principal Value Integrals 38\u003c\/p\u003e \u003cp\u003e1.15 Integrals Involving a Parameter 40\u003c\/p\u003e \u003cp\u003e1.16 Limits of Integration Depending on a Parameter 44\u003c\/p\u003e \u003cp\u003e1.17 Double Integrals 45\u003c\/p\u003e \u003cp\u003e1.18 Properties of Double Integrals 47\u003c\/p\u003e \u003cp\u003e1.19 Triple and Multiple Integrals 48\u003c\/p\u003e \u003cp\u003eReferences 49\u003c\/p\u003e \u003cp\u003eProblems 49\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Vector Analysis 55\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Vector Algebra: Geometric Method 55\u003c\/p\u003e \u003cp\u003e2.1.1 Multiplication of Vectors 57\u003c\/p\u003e \u003cp\u003e2.2 Vector Algebra: Coordinate Representation 60\u003c\/p\u003e \u003cp\u003e2.3 Lines and Planes 65\u003c\/p\u003e \u003cp\u003e2.4 Vector Differential Calculus 67\u003c\/p\u003e \u003cp\u003e2.4.1 Scalar Fields and Vector Fields 67\u003c\/p\u003e \u003cp\u003e2.4.2 Vector Differentiation 69\u003c\/p\u003e \u003cp\u003e2.5 Gradient Operator 70\u003c\/p\u003e \u003cp\u003e2.5.1 Meaning of the Gradient 71\u003c\/p\u003e \u003cp\u003e2.5.2 Directional Derivative 72\u003c\/p\u003e \u003cp\u003e2.6 Divergence and Curl Operators 73\u003c\/p\u003e \u003cp\u003e2.6.1 Meaning of Divergence and the Divergence Theorem 75\u003c\/p\u003e \u003cp\u003e2.7 Vector Integral Calculus in Two Dimensions 79\u003c\/p\u003e \u003cp\u003e2.7.1 Arc Length and Line Integrals 79\u003c\/p\u003e \u003cp\u003e2.7.2 Surface Area and Surface Integrals 83\u003c\/p\u003e \u003cp\u003e2.7.3 An Alternate Way to Write Line Integrals 84\u003c\/p\u003e \u003cp\u003e2.7.4 Green’s Theorem 86\u003c\/p\u003e \u003cp\u003e2.7.5 Interpretations of Green’s Theorem 88\u003c\/p\u003e \u003cp\u003e2.7.6 Extension to Multiply Connected Domains 89\u003c\/p\u003e \u003cp\u003e2.8 Curl Operator and Stokes’s Theorem 92\u003c\/p\u003e \u003cp\u003e2.8.1 On the Plane 92\u003c\/p\u003e \u003cp\u003e2.8.2 In Space 96\u003c\/p\u003e \u003cp\u003e2.8.3 Geometric Interpretation of Curl 99\u003c\/p\u003e \u003cp\u003e2.9 Mixed Operations with the Del Operator 99\u003c\/p\u003e \u003cp\u003e2.10 Potential Theory 102\u003c\/p\u003e \u003cp\u003e2.10.1 Gravitational Field of a Star 105\u003c\/p\u003e \u003cp\u003e2.10.2 Work Done by Gravitational Force 106\u003c\/p\u003e \u003cp\u003e2.10.3 Path Independence and Exact Differentials 108\u003c\/p\u003e \u003cp\u003e2.10.4 Gravity and Conservative Forces 109\u003c\/p\u003e \u003cp\u003e2.10.5 Gravitational Potential 111\u003c\/p\u003e \u003cp\u003e2.10.6 Gravitational Potential Energy of a System 113\u003c\/p\u003e \u003cp\u003e2.10.7 Helmholtz Theorem 115\u003c\/p\u003e \u003cp\u003e2.10.8 Applications of the Helmholtz Theorem 116\u003c\/p\u003e \u003cp\u003e2.10.9 Examples from Physics 120\u003c\/p\u003e \u003cp\u003eReferences 123\u003c\/p\u003e \u003cp\u003eProblems 123\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Generalized Coordinates and Tensors 133\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Transformations between Cartesian Coordinates 134\u003c\/p\u003e \u003cp\u003e3.1.1 Basis Vectors and Direction Cosines 134\u003c\/p\u003e \u003cp\u003e3.1.2 Transformation Matrix and Orthogonality 136\u003c\/p\u003e \u003cp\u003e3.1.3 Inverse Transformation Matrix 137\u003c\/p\u003e \u003cp\u003e3.2 Cartesian Tensors 139\u003c\/p\u003e \u003cp\u003e3.2.1 Algebraic Properties of Tensors 141\u003c\/p\u003e \u003cp\u003e3.2.2 Kronecker Delta and the Permutation Symbol 145\u003c\/p\u003e \u003cp\u003e3.3 Generalized Coordinates 148\u003c\/p\u003e \u003cp\u003e3.3.1 Coordinate Curves and Surfaces 148\u003c\/p\u003e \u003cp\u003e3.3.2 Why Upper and Lower Indices 152\u003c\/p\u003e \u003cp\u003e3.4 General Tensors 153\u003c\/p\u003e \u003cp\u003e3.4.1 Einstein Summation Convention 156\u003c\/p\u003e \u003cp\u003e3.4.2 Line Element 157\u003c\/p\u003e \u003cp\u003e3.4.3 Metric Tensor 157\u003c\/p\u003e \u003cp\u003e3.4.4 How to Raise and Lower Indices 158\u003c\/p\u003e \u003cp\u003e3.4.5 Metric Tensor and the Basis Vectors 160\u003c\/p\u003e \u003cp\u003e3.4.6 Displacement Vector 161\u003c\/p\u003e \u003cp\u003e3.4.7 Line Integrals 162\u003c\/p\u003e \u003cp\u003e3.4.8 Area Element in Generalized Coordinates 164\u003c\/p\u003e \u003cp\u003e3.4.9 Area of a Surface 165\u003c\/p\u003e \u003cp\u003e3.4.10 Volume Element in Generalized Coordinates 169\u003c\/p\u003e \u003cp\u003e3.4.11 Invariance and Covariance 171\u003c\/p\u003e \u003cp\u003e3.5 Differential Operators in Generalized Coordinates 171\u003c\/p\u003e \u003cp\u003e3.5.1 Gradient 171\u003c\/p\u003e \u003cp\u003e3.5.2 Divergence 172\u003c\/p\u003e \u003cp\u003e3.5.3 Curl 174\u003c\/p\u003e \u003cp\u003e3.5.4 Laplacian 178\u003c\/p\u003e \u003cp\u003e3.6 Orthogonal Generalized Coordinates 178\u003c\/p\u003e \u003cp\u003e3.6.1 Cylindrical Coordinates 179\u003c\/p\u003e \u003cp\u003e3.6.2 Spherical Coordinates 184\u003c\/p\u003e \u003cp\u003eReferences 189\u003c\/p\u003e \u003cp\u003eProblems 189\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Determinants and Matrices 197\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Basic Definitions 197\u003c\/p\u003e \u003cp\u003e4.2 Operations with Matrices 198\u003c\/p\u003e \u003cp\u003e4.3 Submatrix and Partitioned Matrices 204\u003c\/p\u003e \u003cp\u003e4.4 Systems of Linear Equations 207\u003c\/p\u003e \u003cp\u003e4.5 Gauss’s Method of Elimination 208\u003c\/p\u003e \u003cp\u003e4.6 Determinants 211\u003c\/p\u003e \u003cp\u003e4.7 Properties of Determinants 214\u003c\/p\u003e \u003cp\u003e4.8 Cramer’s Rule 216\u003c\/p\u003e \u003cp\u003e4.9 Inverse of a Matrix 221\u003c\/p\u003e \u003cp\u003e4.10 Homogeneous Linear Equations 224\u003c\/p\u003e \u003cp\u003eReferences 225\u003c\/p\u003e \u003cp\u003eProblems 225\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Linear Algebra 233\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Fields and Vector Spaces 233\u003c\/p\u003e \u003cp\u003e5.2 Linear Combinations, Generators, and Bases 236\u003c\/p\u003e \u003cp\u003e5.3 Components 238\u003c\/p\u003e \u003cp\u003e5.4 Linear Transformations 241\u003c\/p\u003e \u003cp\u003e5.5 Matrix Representation of Transformations 242\u003c\/p\u003e \u003cp\u003e5.6 Algebra of Transformations 244\u003c\/p\u003e \u003cp\u003e5.7 Change of Basis 246\u003c\/p\u003e \u003cp\u003e5.8 Invariants under Similarity Transformations 247\u003c\/p\u003e \u003cp\u003e5.9 Eigenvalues and Eigenvectors 248\u003c\/p\u003e \u003cp\u003e5.10 Moment of Inertia Tensor 257\u003c\/p\u003e \u003cp\u003e5.11 Inner Product Spaces 262\u003c\/p\u003e \u003cp\u003e5.12 The Inner Product 262\u003c\/p\u003e \u003cp\u003e5.13 Orthogonality and Completeness 265\u003c\/p\u003e \u003cp\u003e5.14 Gram–Schmidt Orthogonalization 267\u003c\/p\u003e \u003cp\u003e5.15 Eigenvalue Problem for Real Symmetric Matrices 268\u003c\/p\u003e \u003cp\u003e5.16 Presence of Degenerate Eigenvalues 270\u003c\/p\u003e \u003cp\u003e5.17 Quadratic Forms 276\u003c\/p\u003e \u003cp\u003e5.18 Hermitian Matrices 279\u003c\/p\u003e \u003cp\u003e5.19 Matrix Representation of Hermitian Operators 283\u003c\/p\u003e \u003cp\u003e5.20 Functions of Matrices 284\u003c\/p\u003e \u003cp\u003e5.21 Function Space and Hilbert Space 286\u003c\/p\u003e \u003cp\u003e5.22 Dirac’s Bra and Ket Vectors 287\u003c\/p\u003e \u003cp\u003eReferences 288\u003c\/p\u003e \u003cp\u003eProblems 289\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Practical Linear Algebra 293\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Systems of Linear Equations 294\u003c\/p\u003e \u003cp\u003e6.1.1 Matrices and Elementary Row Operations 295\u003c\/p\u003e \u003cp\u003e6.1.2 Gauss-Jordan Method 295\u003c\/p\u003e \u003cp\u003e6.1.3 Information From the Row-Echelon Form 300\u003c\/p\u003e \u003cp\u003e6.1.4 Elementary Matrices 301\u003c\/p\u003e \u003cp\u003e6.1.5 Inverse by Gauss-Jordan Row-Reduction 302\u003c\/p\u003e \u003cp\u003e6.1.6 Row Space, Column Space, and Null Space 303\u003c\/p\u003e \u003cp\u003e6.1.7 Bases for Row, Column, and Null Spaces 307\u003c\/p\u003e \u003cp\u003e6.1.8 Vector Spaces Spanned by a Set of Vectors 310\u003c\/p\u003e \u003cp\u003e6.1.9 Rank and Nullity 312\u003c\/p\u003e \u003cp\u003e6.1.10 Linear Transformations 315\u003c\/p\u003e \u003cp\u003e6.2 Numerical Methods of Linear Algebra 317\u003c\/p\u003e \u003cp\u003e6.2.1 Gauss-Jordan Row-Reduction and Partial Pivoting 317\u003c\/p\u003e \u003cp\u003e6.2.2 LU-Factorization 321\u003c\/p\u003e \u003cp\u003e6.2.3 Solutions of Linear Systems by Iteration 325\u003c\/p\u003e \u003cp\u003e6.2.4 Interpolation 328\u003c\/p\u003e \u003cp\u003e6.2.5 Power Method for Eigenvalues 331\u003c\/p\u003e \u003cp\u003e6.2.6 Solution of Equations 333\u003c\/p\u003e \u003cp\u003e6.2.7 Numerical Integration 343\u003c\/p\u003e \u003cp\u003eReferences 349\u003c\/p\u003e \u003cp\u003eProblems 350\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Applications of Linear Algebra 355\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Chemistry and Chemical Engineering 355\u003c\/p\u003e \u003cp\u003e7.1.1 Independent Reactions and Stoichiometric Matrix 356\u003c\/p\u003e \u003cp\u003e7.1.2 Independent Reactions from a Set of Species 359\u003c\/p\u003e \u003cp\u003e7.2 Linear Programming 362\u003c\/p\u003e \u003cp\u003e7.2.1 The Geometric Method 363\u003c\/p\u003e \u003cp\u003e7.2.2 The Simplex Method 367\u003c\/p\u003e \u003cp\u003e7.3 Leontief Input–Output Model of Economy 375\u003c\/p\u003e \u003cp\u003e7.3.1 Leontief Closed Model 375\u003c\/p\u003e \u003cp\u003e7.3.2 Leontief Open Model 378\u003c\/p\u003e \u003cp\u003e7.4 Applications to Geometry 381\u003c\/p\u003e \u003cp\u003e7.4.1 Orbit Calculations 382\u003c\/p\u003e \u003cp\u003e7.5 Elimination Theory 383\u003c\/p\u003e \u003cp\u003e7.5.1 Quadratic Equations and the Resultant 384\u003c\/p\u003e \u003cp\u003e7.6 Coding Theory 388\u003c\/p\u003e \u003cp\u003e7.6.1 Fields and Vector Spaces 388\u003c\/p\u003e \u003cp\u003e7.6.2 Hamming (7,4) Code 390\u003c\/p\u003e \u003cp\u003e7.6.3 Hamming Algorithm for Error Correction 393\u003c\/p\u003e \u003cp\u003e7.7 Cryptography 396\u003c\/p\u003e \u003cp\u003e7.7.1 Single-Key Cryptography 396\u003c\/p\u003e \u003cp\u003e7.8 Graph Theory 399\u003c\/p\u003e \u003cp\u003e7.8.1 Basic Definition 399\u003c\/p\u003e \u003cp\u003e7.8.2 Terminology 400\u003c\/p\u003e \u003cp\u003e7.8.3 Walks, Trails, Paths and Circuits 402\u003c\/p\u003e \u003cp\u003e7.8.4 Trees and Fundamental Circuits 404\u003c\/p\u003e \u003cp\u003e7.8.5 Graph Operations 404\u003c\/p\u003e \u003cp\u003e7.8.6 Cut Sets and Fundamental Cut Sets 405\u003c\/p\u003e \u003cp\u003e7.8.7 Vector Space Associated with a Graph 407\u003c\/p\u003e \u003cp\u003e7.8.8 Rank and Nullity 409\u003c\/p\u003e \u003cp\u003e7.8.9 Subspaces in \u003ci\u003eW\u003csub\u003eG\u003c\/sub\u003e \u003c\/i\u003e410\u003c\/p\u003e \u003cp\u003e7.8.10 Dot Product and Orthogonal vectors 411\u003c\/p\u003e \u003cp\u003e7.8.11 Matrix Representation of Graphs 413\u003c\/p\u003e \u003cp\u003e7.8.12 Dominance Directed Graphs 417\u003c\/p\u003e \u003cp\u003e7.8.13 Gray Codes in Coding Theory 419\u003c\/p\u003e \u003cp\u003eReferences 419\u003c\/p\u003e \u003cp\u003eProblems 420\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Sequences and Series 425\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Sequences 426\u003c\/p\u003e \u003cp\u003e8.2 Infinite Series 430\u003c\/p\u003e \u003cp\u003e8.3 Absolute and Conditional Convergence 431\u003c\/p\u003e \u003cp\u003e8.3.1 Comparison Test 431\u003c\/p\u003e \u003cp\u003e8.3.2 Limit Comparison Test 431\u003c\/p\u003e \u003cp\u003e8.3.3 Integral Test 431\u003c\/p\u003e \u003cp\u003e8.3.4 Ratio Test 432\u003c\/p\u003e \u003cp\u003e8.3.5 Root Test 432\u003c\/p\u003e \u003cp\u003e8.4 Operations with Series 436\u003c\/p\u003e \u003cp\u003e8.5 Sequences and Series of Functions 438\u003c\/p\u003e \u003cp\u003e8.6 \u003ci\u003eM\u003c\/i\u003e-Test for Uniform Convergence 441\u003c\/p\u003e \u003cp\u003e8.7 Properties of Uniformly Convergent Series 441\u003c\/p\u003e \u003cp\u003e8.8 Power Series 443\u003c\/p\u003e \u003cp\u003e8.9 Taylor Series and Maclaurin Series 446\u003c\/p\u003e \u003cp\u003e8.10 Indeterminate Forms and Series 447\u003c\/p\u003e \u003cp\u003eReferences 448\u003c\/p\u003e \u003cp\u003eProblems 448\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Complex Numbers and Functions 453\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 The Algebra of Complex Numbers 454\u003c\/p\u003e \u003cp\u003e9.2 Roots of a Complex Number 458\u003c\/p\u003e \u003cp\u003e9.3 Infinity and the Extended Complex Plane 460\u003c\/p\u003e \u003cp\u003e9.4 Complex Functions 463\u003c\/p\u003e \u003cp\u003e9.5 Limits and Continuity 465\u003c\/p\u003e \u003cp\u003e9.6 Differentiation in the Complex Plane 467\u003c\/p\u003e \u003cp\u003e9.7 Analytic Functions 470\u003c\/p\u003e \u003cp\u003e9.8 Harmonic Functions 471\u003c\/p\u003e \u003cp\u003e9.9 Basic Differentiation Formulas 474\u003c\/p\u003e \u003cp\u003e9.10 Elementary Functions 475\u003c\/p\u003e \u003cp\u003e9.10.1 Polynomials 475\u003c\/p\u003e \u003cp\u003e9.10.2 Exponential Function 476\u003c\/p\u003e \u003cp\u003e9.10.3 Trigonometric Functions 477\u003c\/p\u003e \u003cp\u003e9.10.4 Hyperbolic Functions 478\u003c\/p\u003e \u003cp\u003e9.10.5 Logarithmic Function 479\u003c\/p\u003e \u003cp\u003e9.10.6 Powers of Complex Numbers 481\u003c\/p\u003e \u003cp\u003e9.10.7 Inverse Trigonometric Functions 483\u003c\/p\u003e \u003cp\u003eReferences 483\u003c\/p\u003e \u003cp\u003eProblems 484\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Complex Analysis 491\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Contour Integrals 492\u003c\/p\u003e \u003cp\u003e10.2 Types of Contours 494\u003c\/p\u003e \u003cp\u003e10.3 The Cauchy–Goursat Theorem 497\u003c\/p\u003e \u003cp\u003e10.4 Indefinite Integrals 500\u003c\/p\u003e \u003cp\u003e10.5 Simply and Multiply Connected Domains 502\u003c\/p\u003e \u003cp\u003e10.6 The Cauchy Integral Formula 503\u003c\/p\u003e \u003cp\u003e10.7 Derivatives of Analytic Functions 505\u003c\/p\u003e \u003cp\u003e10.8 Complex Power Series 506\u003c\/p\u003e \u003cp\u003e10.8.1 Taylor Series with the Remainder 506\u003c\/p\u003e \u003cp\u003e10.8.2 Laurent Series with the Remainder 510\u003c\/p\u003e \u003cp\u003e10.9 Convergence of Power Series 514\u003c\/p\u003e \u003cp\u003e10.10 Classification of Singular Points 514\u003c\/p\u003e \u003cp\u003e10.11 Residue Theorem 517\u003c\/p\u003e \u003cp\u003eReferences 522\u003c\/p\u003e \u003cp\u003eProblems 522\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Ordinary Differential Equations 527\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Basic Definitions for Ordinary Differential Equations 528\u003c\/p\u003e \u003cp\u003e11.2 First-Order Differential Equations 530\u003c\/p\u003e \u003cp\u003e11.2.1 Uniqueness of Solution 530\u003c\/p\u003e \u003cp\u003e11.2.2 Methods of Solution 532\u003c\/p\u003e \u003cp\u003e11.2.3 Dependent Variable is Missing 532\u003c\/p\u003e \u003cp\u003e11.2.4 Independent Variable is Missing 532\u003c\/p\u003e \u003cp\u003e11.2.5 The Case of Separable \u003ci\u003ef\u003c\/i\u003e(\u003ci\u003ex, y\u003c\/i\u003e) 532\u003c\/p\u003e \u003cp\u003e11.2.6 Homogeneous \u003ci\u003ef\u003c\/i\u003e(\u003ci\u003ex, y\u003c\/i\u003e) of Zeroth Degree 533\u003c\/p\u003e \u003cp\u003e11.2.7 Solution When \u003ci\u003ef\u003c\/i\u003e(\u003ci\u003ex, y\u003c\/i\u003e) is a Rational Function 533\u003c\/p\u003e \u003cp\u003e11.2.8 Linear Equations of First-order 535\u003c\/p\u003e \u003cp\u003e11.2.9 Exact Equations 537\u003c\/p\u003e \u003cp\u003e11.2.10 Integrating Factors 539\u003c\/p\u003e \u003cp\u003e11.2.11 Bernoulli Equation 542\u003c\/p\u003e \u003cp\u003e11.2.12 Riccati Equation 543\u003c\/p\u003e \u003cp\u003e11.2.13 Equations that Cannot Be Solved for \u003ci\u003ey' \u003c\/i\u003e546\u003c\/p\u003e \u003cp\u003e11.3 Second-Order Differential Equations 548\u003c\/p\u003e \u003cp\u003e11.3.1 The General Case 549\u003c\/p\u003e \u003cp\u003e11.3.2 Linear Homogeneous Equations with Constant Coefficients 551\u003c\/p\u003e \u003cp\u003e11.3.3 Operator Approach 556\u003c\/p\u003e \u003cp\u003e11.3.4 Linear Homogeneous Equations with Variable Coefficients 557\u003c\/p\u003e \u003cp\u003e11.3.5 Cauchy–Euler Equation 560\u003c\/p\u003e \u003cp\u003e11.3.6 Exact Equations and Integrating Factors 561\u003c\/p\u003e \u003cp\u003e11.3.7 Linear Nonhomogeneous Equations 564\u003c\/p\u003e \u003cp\u003e11.3.8 Variation of Parameters 564\u003c\/p\u003e \u003cp\u003e11.3.9 Method of Undetermined Coefficients 566\u003c\/p\u003e \u003cp\u003e11.4 Linear Differential Equations of Higher Order 569\u003c\/p\u003e \u003cp\u003e11.4.1 With Constant Coefficients 569\u003c\/p\u003e \u003cp\u003e11.4.2 With Variable Coefficients 570\u003c\/p\u003e \u003cp\u003e11.4.3 Nonhomogeneous Equations 570\u003c\/p\u003e \u003cp\u003e11.5 Initial Value Problem and Uniqueness of the Solution 571\u003c\/p\u003e \u003cp\u003e11.6 Series Solutions: Frobenius Method 571\u003c\/p\u003e \u003cp\u003e11.6.1 Frobenius Method and First-order Equations 581\u003c\/p\u003e \u003cp\u003eReferences 582\u003c\/p\u003e \u003cp\u003eProblems 582\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Second-Order Differential Equations and Special Functions 589\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Legendre Equation 590\u003c\/p\u003e \u003cp\u003e12.1.1 Series Solution 590\u003c\/p\u003e \u003cp\u003e12.1.2 Effect of Boundary Conditions 593\u003c\/p\u003e \u003cp\u003e12.1.3 Legendre Polynomials 594\u003c\/p\u003e \u003cp\u003e12.1.4 Rodriguez Formula 596\u003c\/p\u003e \u003cp\u003e12.1.5 Generating Function 597\u003c\/p\u003e \u003cp\u003e12.1.6 Special Values 599\u003c\/p\u003e \u003cp\u003e12.1.7 Recursion Relations 600\u003c\/p\u003e \u003cp\u003e12.1.8 Orthogonality 601\u003c\/p\u003e \u003cp\u003e12.1.9 Legendre Series 603\u003c\/p\u003e \u003cp\u003e12.2 Hermite Equation 606\u003c\/p\u003e \u003cp\u003e12.2.1 Series Solution 606\u003c\/p\u003e \u003cp\u003e12.2.2 Hermite Polynomials 610\u003c\/p\u003e \u003cp\u003e12.2.3 Contour Integral Representation 611\u003c\/p\u003e \u003cp\u003e12.2.4 Rodriguez Formula 612\u003c\/p\u003e \u003cp\u003e12.2.5 Generating Function 613\u003c\/p\u003e \u003cp\u003e12.2.6 Special Values 614\u003c\/p\u003e \u003cp\u003e12.2.7 Recursion Relations 614\u003c\/p\u003e \u003cp\u003e12.2.8 Orthogonality 616\u003c\/p\u003e \u003cp\u003e12.2.9 Series Expansions in Hermite Polynomials 618\u003c\/p\u003e \u003cp\u003e12.3 Laguerre Equation 619\u003c\/p\u003e \u003cp\u003e12.3.1 Series Solution 620\u003c\/p\u003e \u003cp\u003e12.3.2 Laguerre Polynomials 621\u003c\/p\u003e \u003cp\u003e12.3.3 Contour Integral Representation 622\u003c\/p\u003e \u003cp\u003e12.3.4 Rodriguez Formula 623\u003c\/p\u003e \u003cp\u003e12.3.5 Generating Function 623\u003c\/p\u003e \u003cp\u003e12.3.6 Special Values and Recursion Relations 624\u003c\/p\u003e \u003cp\u003e12.3.7 Orthogonality 624\u003c\/p\u003e \u003cp\u003e12.3.8 Series Expansions in Laguerre Polynomials 625\u003c\/p\u003e \u003cp\u003eReferences 626\u003c\/p\u003e \u003cp\u003eProblems 626\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Bessel’s Equation and Bessel Functions 629\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Bessel’s Equation and Its Series Solution 630\u003c\/p\u003e \u003cp\u003e13.1.1 Bessel Functions \u003ci\u003eJ\u003csub\u003e±m\u003c\/sub\u003e\u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e)\u003ci\u003e, N\u003csub\u003em\u003c\/sub\u003e\u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e)\u003ci\u003e, \u003c\/i\u003eand \u003ci\u003eH\u003c\/i\u003e\u003csup\u003e(1\u003ci\u003e,\u003c\/i\u003e2)\u003c\/sup\u003e\u003ci\u003e\u003csub\u003em\u003c\/sub\u003e \u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e) 634\u003c\/p\u003e \u003cp\u003e13.1.2 Recursion Relations 639\u003c\/p\u003e \u003cp\u003e13.1.3 Generating Function 639\u003c\/p\u003e \u003cp\u003e13.1.4 Integral Definitions 641\u003c\/p\u003e \u003cp\u003e13.1.5 Linear Independence of Bessel Functions 642\u003c\/p\u003e \u003cp\u003e13.1.6 Modified Bessel Functions \u003ci\u003eI\u003csub\u003em\u003c\/sub\u003e\u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e) and \u003ci\u003eK\u003csub\u003em\u003c\/sub\u003e\u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e) 644\u003c\/p\u003e \u003cp\u003e13.1.7 Spherical Bessel Functions \u003ci\u003ej\u003csub\u003el\u003c\/sub\u003e\u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e)\u003ci\u003e, n\u003csub\u003el\u003c\/sub\u003e\u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e)\u003ci\u003e, \u003c\/i\u003eand \u003ci\u003eh\u003c\/i\u003e\u003csup\u003e(1\u003ci\u003e,\u003c\/i\u003e2)\u003c\/sup\u003e\u003ci\u003e\u003csub\u003el\u003c\/sub\u003e \u003c\/i\u003e(\u003ci\u003ex\u003c\/i\u003e) 645\u003c\/p\u003e \u003cp\u003e13.2 Orthogonality and the Roots of Bessel Functions 648\u003c\/p\u003e \u003cp\u003e13.2.1 Expansion Theorem 652\u003c\/p\u003e \u003cp\u003e13.2.2 Boundary Conditions for the Bessel Functions 652\u003c\/p\u003e \u003cp\u003eReferences 656\u003c\/p\u003e \u003cp\u003eProblems 656\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Partial Differential Equations and Separation of Variables 661\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Separation of Variables in Cartesian Coordinates 662\u003c\/p\u003e \u003cp\u003e14.1.1 Wave Equation 665\u003c\/p\u003e \u003cp\u003e14.1.2 Laplace Equation 666\u003c\/p\u003e \u003cp\u003e14.1.3 Diffusion and Heat Flow Equations 671\u003c\/p\u003e \u003cp\u003e14.2 Separation of Variables in Spherical Coordinates 673\u003c\/p\u003e \u003cp\u003e14.2.1 Laplace Equation 677\u003c\/p\u003e \u003cp\u003e14.2.2 Boundary Conditions for a Spherical Boundary 678\u003c\/p\u003e \u003cp\u003e14.2.3 Helmholtz Equation 682\u003c\/p\u003e \u003cp\u003e14.2.4 Wave Equation 683\u003c\/p\u003e \u003cp\u003e14.2.5 Diffusion and Heat Flow Equations 684\u003c\/p\u003e \u003cp\u003e14.2.6 Time-Independent Schrödinger Equation 685\u003c\/p\u003e \u003cp\u003e14.2.7 Time-Dependent Schrödinger Equation 685\u003c\/p\u003e \u003cp\u003e14.3 Separation of Variables in Cylindrical Coordinates 686\u003c\/p\u003e \u003cp\u003e14.3.1 Laplace Equation 688\u003c\/p\u003e \u003cp\u003e14.3.2 Helmholtz Equation 689\u003c\/p\u003e \u003cp\u003e14.3.3 Wave Equation 690\u003c\/p\u003e \u003cp\u003e14.3.4 Diffusion and Heat Flow Equations 691\u003c\/p\u003e \u003cp\u003eReferences 701\u003c\/p\u003e \u003cp\u003eProblems 701\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Fourier Series 705\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Orthogonal Systems of Functions 705\u003c\/p\u003e \u003cp\u003e15.2 Fourier Series 711\u003c\/p\u003e \u003cp\u003e15.3 Exponential Form of the Fourier Series 712\u003c\/p\u003e \u003cp\u003e15.4 Convergence of Fourier Series 713\u003c\/p\u003e \u003cp\u003e15.5 Sufficient Conditions for Convergence 715\u003c\/p\u003e \u003cp\u003e15.6 The Fundamental Theorem 716\u003c\/p\u003e \u003cp\u003e15.7 Uniqueness of Fourier Series 717\u003c\/p\u003e \u003cp\u003e15.8 Examples of Fourier Series 717\u003c\/p\u003e \u003cp\u003e15.8.1 Square Wave 717\u003c\/p\u003e \u003cp\u003e15.8.2 Triangular Wave 719\u003c\/p\u003e \u003cp\u003e15.8.3 Periodic Extension 720\u003c\/p\u003e \u003cp\u003e15.9 Fourier Sine and Cosine Series 721\u003c\/p\u003e \u003cp\u003e15.10 Change of Interval 722\u003c\/p\u003e \u003cp\u003e15.11 Integration and Differentiation of Fourier Series 723\u003c\/p\u003e \u003cp\u003eReferences 724\u003c\/p\u003e \u003cp\u003eProblems 724\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Fourier and Laplace Transforms 727\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Types of Signals 727\u003c\/p\u003e \u003cp\u003e16.2 Spectral Analysis and Fourier Transforms 730\u003c\/p\u003e \u003cp\u003e16.3 Correlation with Cosines and Sines 731\u003c\/p\u003e \u003cp\u003e16.4 Correlation Functions and Fourier Transforms 735\u003c\/p\u003e \u003cp\u003e16.5 Inverse Fourier Transform 736\u003c\/p\u003e \u003cp\u003e16.6 Frequency Spectrums 736\u003c\/p\u003e \u003cp\u003e16.7 Dirac-Delta Function 738\u003c\/p\u003e \u003cp\u003e16.8 A Case with Two Cosines 739\u003c\/p\u003e \u003cp\u003e16.9 General Fourier Transforms and Their Properties 740\u003c\/p\u003e \u003cp\u003e16.10 Basic Definition of Laplace Transform 743\u003c\/p\u003e \u003cp\u003e16.11 Differential Equations and Laplace Transforms 746\u003c\/p\u003e \u003cp\u003e16.12 Transfer Functions and Signal Processors 748\u003c\/p\u003e \u003cp\u003e16.13 Connection of Signal Processors 750\u003c\/p\u003e \u003cp\u003eReferences 753\u003c\/p\u003e \u003cp\u003eProblems 753\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Calculus of Variations 757\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 A Simple Case 758\u003c\/p\u003e \u003cp\u003e17.2 Variational Analysis 759\u003c\/p\u003e \u003cp\u003e17.2.1 Case I: The Desired Function is Prescribed at the End Points 761\u003c\/p\u003e \u003cp\u003e17.2.2 Case II: Natural Boundary Conditions 762\u003c\/p\u003e \u003cp\u003e17.3 Alternate Form of Euler Equation 763\u003c\/p\u003e \u003cp\u003e17.4 Variational Notation 765\u003c\/p\u003e \u003cp\u003e17.5 A More General Case 767\u003c\/p\u003e \u003cp\u003e17.6 Hamilton’s Principle 772\u003c\/p\u003e \u003cp\u003e17.7 Lagrange’s Equations of Motion 773\u003c\/p\u003e \u003cp\u003e17.8 Definition of Lagrangian 777\u003c\/p\u003e \u003cp\u003e17.9 Presence of Constraints in Dynamical Systems 779\u003c\/p\u003e \u003cp\u003e17.10 Conservation Laws 783\u003c\/p\u003e \u003cp\u003eReferences 784\u003c\/p\u003e \u003cp\u003eProblems 784\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Probability Theory and Distributions 789\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Introduction to Probability Theory 790\u003c\/p\u003e \u003cp\u003e18.1.1 Fundamental Concepts 790\u003c\/p\u003e \u003cp\u003e18.1.2 Basic Axioms of Probability 791\u003c\/p\u003e \u003cp\u003e18.1.3 Basic Theorems of Probability 791\u003c\/p\u003e \u003cp\u003e18.1.4 Statistical Definition of Probability 794\u003c\/p\u003e \u003cp\u003e18.1.5 Conditional Probability and Multiplication Theorem 795\u003c\/p\u003e \u003cp\u003e18.1.6 Bayes’ Theorem 796\u003c\/p\u003e \u003cp\u003e18.1.7 Geometric Probability and Buffon’s Needle Problem 798\u003c\/p\u003e \u003cp\u003e18.2 Permutations and Combinations 800\u003c\/p\u003e \u003cp\u003e18.2.1 The Case of Distinguishable Balls with Replacement 800\u003c\/p\u003e \u003cp\u003e18.2.2 The Case of Distinguishable Balls without Replacement 801\u003c\/p\u003e \u003cp\u003e18.2.3 The Case of Indistinguishable Balls 802\u003c\/p\u003e \u003cp\u003e18.2.4 Binomial and Multinomial Coefficients 803\u003c\/p\u003e \u003cp\u003e18.3 Applications to Statistical Mechanics 804\u003c\/p\u003e \u003cp\u003e18.3.1 Boltzmann Distribution for Solids 805\u003c\/p\u003e \u003cp\u003e18.3.2 Boltzmann Distribution for Gases 807\u003c\/p\u003e \u003cp\u003e18.3.3 Bose–Einstein Distribution for Perfect Gases 808\u003c\/p\u003e \u003cp\u003e18.3.4 Fermi–Dirac Distribution 810\u003c\/p\u003e \u003cp\u003e18.4 Statistical Mechanics and Thermodynamics 811\u003c\/p\u003e \u003cp\u003e18.4.1 Probability and Entropy 811\u003c\/p\u003e \u003cp\u003e18.4.2 Derivation of \u003ci\u003eβ \u003c\/i\u003e812\u003c\/p\u003e \u003cp\u003e18.5 Random Variables and Distributions 814\u003c\/p\u003e \u003cp\u003e18.6 Distribution Functions and Probability 817\u003c\/p\u003e \u003cp\u003e18.7 Examples of Continuous Distributions 819\u003c\/p\u003e \u003cp\u003e18.7.1 Uniform Distribution 819\u003c\/p\u003e \u003cp\u003e18.7.2 Gaussian or Normal Distribution 820\u003c\/p\u003e \u003cp\u003e18.7.3 Gamma Distribution 821\u003c\/p\u003e \u003cp\u003e18.8 Discrete Probability Distributions 821\u003c\/p\u003e \u003cp\u003e18.8.1 Uniform Distribution 822\u003c\/p\u003e \u003cp\u003e18.8.2 Binomial Distribution 822\u003c\/p\u003e \u003cp\u003e18.8.3 Poisson Distribution 824\u003c\/p\u003e \u003cp\u003e18.9 Fundamental Theorem of Averages 825\u003c\/p\u003e \u003cp\u003e18.10 Moments of Distribution Functions 826\u003c\/p\u003e \u003cp\u003e18.10.1 Moments of the Gaussian Distribution 827\u003c\/p\u003e \u003cp\u003e18.10.2 Moments of the Binomial Distribution 827\u003c\/p\u003e \u003cp\u003e18.10.3 Moments of the Poisson Distribution 829\u003c\/p\u003e \u003cp\u003e18.11 Chebyshev’s Theorem 831\u003c\/p\u003e \u003cp\u003e18.12 Law of Large Numbers 832\u003c\/p\u003e \u003cp\u003eReferences 833\u003c\/p\u003e \u003cp\u003eProblems 834\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 Information Theory 841\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 Elements of Information Processing Mechanisms 844\u003c\/p\u003e \u003cp\u003e19.2 Classical Information Theory 846\u003c\/p\u003e \u003cp\u003e19.2.1 Prior Uncertainty and Entropy of Information 848\u003c\/p\u003e \u003cp\u003e19.2.2 Joint and Conditional Entropies of Information 851\u003c\/p\u003e \u003cp\u003e19.2.3 Decision Theory 854\u003c\/p\u003e \u003cp\u003e19.2.4 Decision Theory and Game Theory 856\u003c\/p\u003e \u003cp\u003e19.2.5 Traveler’s Dilemma and Nash Equilibrium 862\u003c\/p\u003e \u003cp\u003e19.2.6 Classical Bit or Cbit 866\u003c\/p\u003e \u003cp\u003e19.2.7 Operations on Cbits 869\u003c\/p\u003e \u003cp\u003e19.3 Quantum Information Theory 871\u003c\/p\u003e \u003cp\u003e19.3.1 Basic Quantum Theory 872\u003c\/p\u003e \u003cp\u003e19.3.2 Single-Particle Systems and Quantum Information 878\u003c\/p\u003e \u003cp\u003e19.3.3 Mach–Zehnder Interferometer 880\u003c\/p\u003e \u003cp\u003e19.3.4 Mathematics of the Mach–Zehnder Interferometer 882\u003c\/p\u003e \u003cp\u003e19.3.5 Quantum Bit or Qbit 886\u003c\/p\u003e \u003cp\u003e19.3.6 The No-Cloning Theorem 889\u003c\/p\u003e \u003cp\u003e19.3.7 Entanglement and Bell States 890\u003c\/p\u003e \u003cp\u003e19.3.8 Quantum Dense Coding 895\u003c\/p\u003e \u003cp\u003e19.3.9 Quantum Teleportation 896\u003c\/p\u003e \u003cp\u003eReferences 900\u003c\/p\u003e \u003cp\u003eProblems 901\u003c\/p\u003e \u003cp\u003eFurther Reading 907\u003c\/p\u003e \u003cp\u003eIndex 915\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49407089770839,"sku":"9781119580249","price":124.4,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781119580249.jpg?v=1730498142","url":"https:\/\/bookcurl.com\/products\/essentials-of-mathematical-methods-in-science-and-engineering-9781119580249","provider":"Book Curl","version":"1.0","type":"link"}