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Solution Manual for Advanced Engineering Mathematics (10th Edition) by Erwin Kreyszig

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  • ISBN-10:  0470458364 / ISBN-13:  9780470458365
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    • Edition: 10th Edition
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    • Resource Type : Solution Manual
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    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 1 Review Questions and Problems 43
    Summary of Chapter 1 44
    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 2 Review Questions and Problems 102
    Summary of Chapter 2 103
    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 3 Review Questions and Problems 122
    Summary of Chapter 3 123
    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 4 Review Questions and Problems 164
    Summary of Chapter 4 165
    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 Jv(x) 187
    5.5 Bessel Functions of the Yv(x). General Solution 196
    Chapter 5 Review Questions and Problems 200
    Summary of Chapter 5 201
    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
    Chapter 6 Review Questions and Problems 251
    Summary of Chapter 6 253
    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 7 Review Questions and Problems 318
    Summary of Chapter 7 320
    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 8 Review Questions and Problems 352
    Summary of Chapter 8 353
    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 9 Review Questions and Problems 409
    Summary of Chapter 9 410
    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
    Chapter 10 Review Questions and Problems 469
    Summary of Chapter 10 470
    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 11 Review Questions and Problems 537
    Summary of Chapter 11 538
    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
    Chapter 12 Review Questions and Problems 603
    Summary of Chapter 12 604
    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 13 Review Questions and Problems 641
    Summary of Chapter 13 641
    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 14 Review Questions and Problems 668
    Summary of Chapter 14 669
    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 15 Review Questions and Problems 706
    Summary of Chapter 15 706
    Chapter 16 Laurent Series. Residue Integration 708
    16.1 Laurent Series 708
    16.2 Singularities and Zeros. Infinity 715
    16.3 Residue Integration Method 719
    16.4 Residue Integration of Real Integrals 725
    Chapter 16 Review Questions and Problems 733
    Summary of Chapter 16 734
    Chapter 17 Conformal Mapping 736
    17.1 Geometry of Analytic Functions: Conformal Mapping 737
    17.2 Linear Fractional Transformations (Möbius Transformations) 742
    17.3 Special Linear Fractional Transformations 746
    17.4 Conformal Mapping by Other Functions 750
    17.5 Riemann Surfaces. Optional 754
    Chapter 17 Review Questions and Problems 756
    Summary of Chapter 17 757
    Chapter 18 Complex Analysis and Potential Theory 758
    18.1 Electrostatic Fields 759
    18.2 Use of Conformal Mapping. Modeling 763
    18.3 Heat Problems 767
    18.4 Fluid Flow 771
    18.5 Poisson’s Integral Formula for Potentials 777
    18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem 781
    Chapter 18 Review Questions and Problems 785
    Summary of Chapter 18 786
    Part E Numeric Analysis 787
    Software 788
    Chapter 19 Numerics in General 790
    19.1 Introduction 790
    19.2 Solution of Equations by Iteration 798
    19.3 Interpolation 808
    19.4 Spline Interpolation 820
    19.5 Numeric Integration and Differentiation 827
    Chapter 19 Review Questions and Problems 841
    Summary of Chapter 19 842
    Chapter 20 Numeric Linear Algebra 844
    20.1 Linear Systems: Gauss Elimination 844
    20.2 Linear Systems: LU-Factorization, Matrix Inversion 852
    20.3 Linear Systems: Solution by Iteration 858
    20.4 Linear Systems: Ill-Conditioning, Norms 864
    20.5 Least Squares Method 872
    20.6 Matrix Eigenvalue Problems: Introduction 876
    20.7 Inclusion of Matrix Eigenvalues 879
    20.8 Power Method for Eigenvalues 885
    20.9 Tridiagonalization and QR-Factorization 888
    Chapter 20 Review Questions and Problems 896
    Summary of Chapter 20 898
    Chapter 21 Numerics for ODEs and PDEs 900
    21.1 Methods for First-Order ODEs 901
    21.2 Multistep Methods 911
    21.3 Methods for Systems and Higher Order ODEs 915
    21.4 Methods for Elliptic PDEs 922
    21.5 Neumann and Mixed Problems. Irregular Boundary 931
    21.6 Methods for Parabolic PDEs 936
    21.7 Method for Hyperbolic PDEs 942
    Chapter 21 Review Questions and Problems 945
    Summary of Chapter 21 946
    Part F Optimization, Graphs 949
    Chapter 22 Unconstrained Optimization. Linear Programming 950
    22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 951
    22.2 Linear Programming 954
    22.3 Simplex Method 958
    22.4 Simplex Method: Difficulties 962
    Chapter 22 Review Questions and Problems 968
    Summary of Chapter 22 969
    Chapter 23 Graphs. Combinatorial Optimization 970
    23.1 Graphs and Digraphs 970
    23.2 Shortest Path Problems. Complexity 975
    23.3 Bellman’s Principle. Dijkstra’s Algorithm 980
    23.4 Shortest Spanning Trees: Greedy Algorithm 984
    23.5 Shortest Spanning Trees: Prim’s Algorithm 988
    23.6 Flows in Networks 991
    23.7 Maximum Flow: Ford–Fulkerson Algorithm 998
    23.8 Bipartite Graphs. Assignment Problems 1001
    Chapter 23 Review Questions and Problems 1006
    Summary of Chapter 23 1007
    Part G Probability, Statistics 1009
    Software 1009
    Chapter 24 Data Analysis. Probability Theory 1011
    24.1 Data Representation. Average. Spread 1011
    24.2 Experiments, Outcomes, Events 1015
    24.3 Probability 1018
    24.4 Permutations and Combinations 1024
    24.5 Random Variables. Probability Distributions 1029
    24.6 Mean and Variance of a Distribution 1035
    24.7 Binomial, Poisson, and Hypergeometric Distributions 1039
    24.8 Normal Distribution 1045
    24.9 Distributions of Several Random Variables 1051
    Chapter 24 Review Questions and Problems 1060
    Summary of Chapter 24 1060
    Chapter 25 Mathematical Statistics 1063
    25.1 Introduction. Random Sampling 1063
    25.2 Point Estimation of Parameters 1065
    25.3 Confidence Intervals 1068
    25.4 Testing Hypotheses. Decisions 1077
    25.5 Quality Control 1087
    25.6 Acceptance Sampling 1092
    25.7 Goodness of Fit. χ 2-Test 1096
    25.8 Nonparametric Tests 1100
    25.9 Regression. Fitting Straight Lines. Correlation 1103
    Chapter 25 Review Questions and Problems 1111
    Summary of Chapter 25 1112
    Appendix 1 References A1
    Appendix 2 Answers to Odd-Numbered Problems A4
    Appendix 3 Auxiliary Material A63
    A3.1 Formulas for Special Functions A63
    A3.2 Partial Derivatives A69
    A3.3 Sequences and Series A72
    A3.4 Grad, Div, Curl, ∇2 in Curvilinear Coordinates A74
    Appendix 4 Additional Proofs A77
    Appendix 5 Tables A97
    Index I1
    Photo Credits P1

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