Advanced Engineering Mathematics, International Student Version

This market-leading text is known for its comprehensive coverage, careful and correct mathematics, outstanding exercises, and self contained subject matter parts for maximum flexibility. The new edition continues with the tradition of providing instructors and students with a comprehensive and up-to-date resource for teaching and learning engineering mathematics, that is, applied mathematics for engineers and physicists, mathematicians and computer scientists, as well as members of other disciplines. This edition can be accompanied with WileyPLUS 5.0, a powerful online teaching and learning environment that integrates the entire digital textbook with the most effective resources to fit every learning style.

Table of Contents:
PART A Ordinary Differential Equations (ODEs) 1 CHAPTER 1 First-Order ODEs 2 1.1 Basic Concepts. Modeling 2 1.2 Geometric Meaning of y ƒ(x, y). Direction Fields, Euler’s Method 9 1.3 Separable ODEs. Modeling 12 1.4 Exact ODEs. Integrating Factors 20 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 27 1.6 Orthogonal Trajectories. Optional 36 1.7 Existence and Uniqueness of Solutions for Initial Value Problems 38 CHAPTER 2 Second-Order Linear ODEs 46 2.1 Homogeneous Linear ODEs of Second Order 46 2.2 Homogeneous Linear ODEs with Constant Coefficients 53 2.3 Differential Operators. Optional 60 2.4 Modeling of Free Oscillations of a Mass–Spring System 62 2.5 Euler–Cauchy Equations 71 2.6 Existence and Uniqueness of Solutions. Wronskian 74 2.7 Nonhomogeneous ODEs 79 2.8 Modeling: Forced Oscillations. Resonance 85 2.9 Modeling: Electric Circuits 93 2.10 Solution by Variation of Parameters 99 CHAPTER 3 Higher Order Linear ODEs 105 3.1 Homogeneous Linear ODEs 105 3.2 Homogeneous Linear ODEs with Constant Coefficients 111 3.3 Nonhomogeneous Linear ODEs 116 CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods 124 4.0 For Reference: Basics of Matrices and Vectors 124 4.1 Systems of ODEs as Models in Engineering Applications 130 4.2 Basic Theory of Systems of ODEs. Wronskian 137 4.3 Constant-Coefficient Systems. Phase Plane Method 140 4.4 Criteria for Critical Points. Stability 148 4.5 Qualitative Methods for Nonlinear Systems 152 4.6 Nonhomogeneous Linear Systems of ODEs 160 CHAPTER 5 Series Solutions of ODEs. Special Functions 167 5.1 Power Series Method 167 5.2 Legendre's Equation. Legendre Polynomials Pn(x) 175 5.3 Extended Power Series Method: Frobenius Method 180 5.4 Bessel’s Equation. Bessel Functions (x) 187 5.5 Bessel Functions of the Y (x). General Solution 196 CHAPTER 6 Laplace Transforms 203 6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) 204 6.2 Transforms of Derivatives and Integrals. ODEs 211 6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting) 217 6.4 Short Impulses. Dirac's Delta Function. Partial Fractions 225 6.5 Convolution. Integral Equations 232 6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients 238 6.7 Systems of ODEs 242 6.8 Laplace Transform: General Formulas 248 6.9 Table of Laplace Transforms 249 PART B Linear Algebra. Vector Calculus 255 CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems 256 7.1 Matrices, Vectors: Addition and Scalar Multiplication 257 7.2 Matrix Multiplication 263 7.3 Linear Systems of Equations. Gauss Elimination 272 7.4 Linear Independence. Rank of a Matrix. Vector Space 282 7.5 Solutions of Linear Systems: Existence, Uniqueness 288 7.6 For Reference: Second- and Third-Order Determinants 291 7.7 Determinants. Cramer’s Rule 293 7.8 Inverse of a Matrix. Gauss–Jordan Elimination 301 7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional 309 CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems 322 8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors 323 8.2 Some Applications of Eigenvalue Problems 329 8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices 334 8.4 Eigenbases. Diagonalization. Quadratic Forms 339 8.5 Complex Matrices and Forms. Optional 346 CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl 354 9.1 Vectors in 2-Space and 3-Space 354 9.2 Inner Product (Dot Product) 361 9.3 Vector Product (Cross Product) 368 9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives 375 9.5 Curves. Arc Length. Curvature. Torsion 381 9.6 Calculus Review: Functions of Several Variables. Optional 392 9.7 Gradient of a Scalar Field. Directional Derivative 395 9.8 Divergence of a Vector Field 403 9.9 Curl of a Vector Field 406 CHAPTER 10 Vector Integral Calculus. Integral Theorems 413 10.1 Line Integrals 413 10.2 Path Independence of Line Integrals 419 10.3 Calculus Review: Double Integrals. Optional 426 10.4 Green’s Theorem in the Plane 433 10.5 Surfaces for Surface Integrals 439 10.6 Surface Integrals 443 10.7 Triple Integrals. Divergence Theorem of Gauss 452 10.8 Further Applications of the Divergence Theorem 458 10.9 Stokes’s Theorem 463 PART C Fourier Analysis. Partial Differential Equations (PDEs) 473 CHAPTER 11 Fourier Analysis 474 11.1 Fourier Series 474 11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 483 11.3 Forced Oscillations 492 11.4 Approximation by Trigonometric Polynomials 495 11.5 Sturm–Liouville Problems. Orthogonal Functions 498 11.6 Orthogonal Series. Generalized Fourier Series 504 11.7 Fourier Integral 510 11.8 Fourier Cosine and Sine Transforms 518 11.9 Fourier Transform. Discrete and Fast Fourier Transforms 522 11.10 Tables of Transforms 534 CHAPTER 12 Partial Differential Equations (PDEs) 540 12.1 Basic Concepts of PDEs 540 12.2 Modeling: Vibrating String, Wave Equation 543 12.3 Solution by Separating Variables. Use of Fourier Series 545 12.4 D’Alembert’s Solution of the Wave Equation. Characteristics 553 12.5 Modeling: Heat Flow from a Body in Space. Heat Equation 557 12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem 558 12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms 568 12.8 Modeling: Membrane, Two-Dimensional Wave Equation 575 12.9 Rectangular Membrane. Double Fourier Series 577 12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series 585 12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential 593 12.12 Solution of PDEs by Laplace Transforms 600 PART D Complex Analysis 607 CHAPTER 13 Complex Numbers and Functions. Complex Differentiation 608 13.1 Complex Numbers and Their Geometric Representation 608 13.2 Polar Form of Complex Numbers. Powers and Roots 613 13.3 Derivative. Analytic Function 619 13.4 Cauchy–Riemann Equations. Laplace’s Equation 625 13.5 Exponential Function 630 13.6 Trigonometric and Hyperbolic Functions. Euler's Formula 633 13.7 Logarithm. General Power. Principal Value 636 CHAPTER 14 Complex Integration 643 14.1 Line Integral in the Complex Plane 643 14.2 Cauchy's Integral Theorem 652 14.3 Cauchy's Integral Formula 660 14.4 Derivatives of Analytic Functions 664 CHAPTER 15 Power Series, Taylor Series 671 15.1 Sequences, Series, Convergence Tests 671 15.2 Power Series 680 15.3 Functions Given by Power Series 685 15.4 Taylor and Maclaurin Series 690 15.5 Uniform Convergence. Optional 698 CHAPTER 16 Laurent Series. Residue Integration 708 16.1 Laurent Series 708 16.2 Singularities and Zeros. Infinity 714 16.3 Residue Integration Method 719 16.4 Residue Integration of Real Integrals 725 CHAPTER 17 Conformal Mapping 735 17.1 Geometry of Analytic Functions: Conformal Mapping 736 17.2 Linear Fractional Transformations (Möbius Transformations) 741 17.3 Special Linear Fractional Transformations 745 17.4 Conformal Mapping by Other Functions 749 17.5 Riemann Surfaces. Optional 753 CHAPTER 18 Complex Analysis and Potential Theory 756 18.1 Electrostatic Fields 757 18.2 Use of Conformal Mapping. Modeling 761 18.3 Heat Problems 765 18.4 Fluid Flow 768 18.5 Poisson's Integral Formula for Potentials 774 18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem 778 PART E Numeric Analysis 785 Software 786 CHAPTER 19 Numerics in General 788 19.1 Introduction 788 19.2 Solution of Equations by Iteration 795 19.3 Interpolation 805 19.4 Spline Interpolation 817 19.5 Numeric Integration and Differentiation 824 CHAPTER 20 Numeric Linear Algebra 841 20.1 Linear Systems: Gauss Elimination 841 20.2 Linear Systems: LU-Factorization, Matrix Inversion 849 20.3 Linear Systems: Solution by Iteration 855 20.4 Linear Systems: Ill-Conditioning, Norms 861 20.5 Least Squares Method 869 20.6 Matrix Eigenvalue Problems: Introduction 873 20.7 Inclusion of Matrix Eigenvalues 876 20.8 Power Method for Eigenvalues 882 20.9 Tridiagonalization and QR-Factorization 885 CHAPTER 21 Numerics for ODEs and PDEs 897 21.1 Methods for First-Order ODEs 898 21.2 Multistep Methods 908 21.3 Methods for Systems and Higher Order ODEs 912 21.4 Methods for Elliptic PDEs 919 21.5 Neumann and Mixed Problems. Irregular Boundary 928 21.6 Methods for Parabolic PDEs 933 21.7 Method for Hyperbolic PDEs 939 PART F Optimization, Graphs 947 CHAPTER 22 Unconstrained Optimization. Linear Programming 948 22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 949 22.2 Linear Programming 952 22.3 Simplex Method 956 22.4 Simplex Method: Difficulties 960 CHAPTER 23 Graphs. Combinatorial Optimization 967 23.1 Graphs and Digraphs 967 23.2 Shortest Path Problems. Complexity 972 23.3 Bellman's Principle. Dijkstra’s Algorithm 977 23.4 Shortest Spanning Trees: Greedy Algorithm 980 23.5 Shortest Spanning Trees: Prim’s Algorithm 984 23.6 Flows in Networks 987 23.7 Maximum Flow: Ford–Fulkerson Algorithm 993 23.8 Bipartite Graphs. Assignment Problems 996 APPENDIX 1 References A1 APPENDIX 2 Answers to Selected Problems A4 APPENDIX 3 Auxiliary Material A51 A3.1 Formulas for Special Functions A51 A3.2 Partial Derivatives A57 A3.3 Sequences and Series A60 A3.4 Grad, Div, Curl, 2 in Curvilinear Coordinates A62 APPENDIX 4 Additional Proofs A65 APPENDIX 5 Tables A85 INDEX I1 PHOTO CREDITS P1