Advanced Engineering Mathematics by Erwin Kreyszig 10th Edition
Publisher: Wiley, Year: 2011
Pages: 1283
Size: 21 mb
ISBN: 0470458364,9780470458365
The tenth edition of this bestselling text includes examples in more detail and more applied exercises; both changes are aimed at making the material more relevant and accessible to readers. Kreyszig introduces engineers and computer scientists to advanced math topics as they relate to practical problems. It goes into the following topics at great depth differential equations, partial differential equations, Fourier analysis, vector analysis, complex analysis, and linear algebra/differential equations.
Contents:
P A R T A Ordinary Differential Equations (ODEs)
CHAPTER 1 First-Order ODEs
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 462.2 Homogeneous Linear ODEs with Constant Coefficients 532.3 Differential Operators. Optional 602.4 Modeling of Free Oscillations of a Mass–Spring System 622.5 Euler–Cauchy Equations 712.6 Existence and Uniqueness of Solutions. Wronskian 742.7 Nonhomogeneous ODEs 792.8 Modeling: Forced Oscillations. Resonance 852.9 Modeling: Electric Circuits 932.10 Solution by Variation of Parameters 99Chapter 2 Review Questions and Problems 102Summary of Chapter 2 103
CHAPTER 3 Higher Order Linear ODEs 105
3.1 Homogeneous Linear ODEs 1053.2 Homogeneous Linear ODEs with Constant Coefficients 1113.3 Nonhomogeneous Linear ODEs 116Chapter 3 Review Questions and Problems 122Summary of Chapter 3 123
CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods 124
4.0 For Reference: Basics of Matrices and Vectors 1244.1 Systems of ODEs as Models in Engineering Applications 1304.2 Basic Theory of Systems of ODEs. Wronskian 1374.3 Constant-Coefficient Systems. Phase Plane Method 1404.4 Criteria for Critical Points. Stability 1484.5 Qualitative Methods for Nonlinear Systems 1524.6 Nonhomogeneous Linear Systems of ODEs 160Chapter 4 Review Questions and Problems 164Summary of Chapter 4 165
CHAPTER 5 Series Solutions of ODEs. Special Functions 167
5.1 Power Series Method 1675.2 Legendre’s Equation. Legendre Polynomials Pn(x) 1755.3 Extended Power Series Method: Frobenius Method 1805.4 Bessel’s Equation. Bessel Functions J(x) 1875.5 Bessel Functions of the Y(x). General Solution 196Chapter 5 Review Questions and Problems 200Summary of Chapter 5 201
CHAPTER 6 Laplace Transforms 203
6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) 2046.2 Transforms of Derivatives and Integrals. ODEs 2116.3 Unit Step Function (Heaviside Function).Second Shifting Theorem (t-Shifting) 2176.4 Short Impulses. Dirac’s Delta Function. Partial Fractions 2256.5 Convolution. Integral Equations 2326.6 Differentiation and Integration of Transforms.ODEs with Variable Coefficients 2386.7 Systems of ODEs 2426.8 Laplace Transform: General Formulas 2486.9 Table of Laplace Transforms 249Chapter 6 Review Questions and Problems 251Summary of Chapter 6 253
P A R T B Linear Algebra. Vector Calculus 255
CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems 256
7.1 Matrices, Vectors: Addition and Scalar Multiplication 2577.2 Matrix Multiplication 2637.3 Linear Systems of Equations. Gauss Elimination 2727.4 Linear Independence. Rank of a Matrix. Vector Space 2827.5 Solutions of Linear Systems: Existence, Uniqueness 2887.6 For Reference: Second- and Third-Order Determinants 2917.7 Determinants. Cramer’s Rule 2937.8 Inverse of a Matrix. Gauss–Jordan Elimination 3017.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional 309Chapter 7 Review Questions and Problems 318Summary of Chapter 7 320
CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems 322
8.1 The Matrix Eigenvalue Problem.Determining Eigenvalues and Eigenvectors 3238.2 Some Applications of Eigenvalue Problems 3298.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices 3348.4 Eigenbases. Diagonalization. Quadratic Forms 3398.5 Complex Matrices and Forms. Optional 346Chapter 8 Review Questions and Problems 352Summary of Chapter 8 353xvi Contents
CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl 354
9.1 Vectors in 2-Space and 3-Space 3549.2 Inner Product (Dot Product) 3619.3 Vector Product (Cross Product) 3689.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives 3759.5 Curves. Arc Length. Curvature. Torsion 3819.6 Calculus Review: Functions of Several Variables. Optional 3929.7 Gradient of a Scalar Field. Directional Derivative 3959.8 Divergence of a Vector Field 4029.9 Curl of a Vector Field 406Chapter 9 Review Questions and Problems 409Summary of Chapter 9 410
CHAPTER 10 Vector Integral Calculus. Integral Theorems 413
10.1 Line Integrals 41310.2 Path Independence of Line Integrals 41910.3 Calculus Review: Double Integrals. Optional 42610.4 Green’s Theorem in the Plane 43310.5 Surfaces for Surface Integrals 43910.6 Surface Integrals 44310.7 Triple Integrals. Divergence Theorem of Gauss 45210.8 Further Applications of the Divergence Theorem 45810.9 Stokes’s Theorem 463Chapter 10 Review Questions and Problems 469Summary of Chapter 10 470
P A R T C Fourier Analysis. Partial Differential Equations (PDEs) 473
CHAPTER 11 Fourier Analysis 474
11.1 Fourier Series 47411.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 48311.3 Forced Oscillations 49211.4 Approximation by Trigonometric Polynomials 49511.5 Sturm–Liouville Problems. Orthogonal Functions 49811.6 Orthogonal Series. Generalized Fourier Series 50411.7 Fourier Integral 51011.8 Fourier Cosine and Sine Transforms 51811.9 Fourier Transform. Discrete and Fast Fourier Transforms 52211.10 Tables of Transforms 534Chapter 11 Review Questions and Problems 537Summary of Chapter 11 538
CHAPTER 12 Partial Differential Equations (PDEs) 540
12.1 Basic Concepts of PDEs 54012.2 Modeling: Vibrating String, Wave Equation 54312.3 Solution by Separating Variables. Use of Fourier Series 54512.4 D’Alembert’s Solution of the Wave Equation. Characteristics 55312.5 Modeling: Heat Flow from a Body in Space. Heat Equation 557Contents xvii12.6 Heat Equation: Solution by Fourier Series.Steady Two-Dimensional Heat Problems. Dirichlet Problem 55812.7 Heat Equation: Modeling Very Long Bars.Solution by Fourier Integrals and Transforms 56812.8 Modeling: Membrane, Two-Dimensional Wave Equation 57512.9 Rectangular Membrane. Double Fourier Series 57712.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series 58512.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential 59312.12 Solution of PDEs by Laplace Transforms 600Chapter 12 Review Questions and Problems 603Summary of Chapter 12 604
P A R T D Complex Analysis 607
CHAPTER 13 Complex Numbers and Functions. Complex Differentiation 608
13.1 Complex Numbers and Their Geometric Representation 60813.2 Polar Form of Complex Numbers. Powers and Roots 61313.3 Derivative. Analytic Function 61913.4 Cauchy–Riemann Equations. Laplace’s Equation 62513.5 Exponential Function 63013.6 Trigonometric and Hyperbolic Functions. Euler’s Formula 63313.7 Logarithm. General Power. Principal Value 636Chapter 13 Review Questions and Problems 641Summary of Chapter 13 641
CHAPTER 14 Complex Integration 643
14.1 Line Integral in the Complex Plane 64314.2 Cauchy’s Integral Theorem 65214.3 Cauchy’s Integral Formula 66014.4 Derivatives of Analytic Functions 664Chapter 14 Review Questions and Problems 668Summary of Chapter 14 669
CHAPTER 15 Power Series, Taylor Series 671
15.1 Sequences, Series, Convergence Tests 67115.2 Power Series 68015.3 Functions Given by Power Series 68515.4 Taylor and Maclaurin Series 69015.5 Uniform Convergence. Optional 698Chapter 15 Review Questions and Problems 706Summary of Chapter 15 706
CHAPTER 16 Laurent Series. Residue Integration 708
16.1 Laurent Series 70816.2 Singularities and Zeros. Infinity 71516.3 Residue Integration Method 71916.4 Residue Integration of Real Integrals 725Chapter 16 Review Questions and Problems 733Summary of Chapter 16 734xviii Contents
CHAPTER 17 Conformal Mapping 736
17.1 Geometry of Analytic Functions: Conformal Mapping 73717.2 Linear Fractional Transformations (Möbius Transformations) 74217.3 Special Linear Fractional Transformations 74617.4 Conformal Mapping by Other Functions 75017.5 Riemann Surfaces. Optional 754Chapter 17 Review Questions and Problems 756Summary of Chapter 17 757
CHAPTER 18 Complex Analysis and Potential Theory 758
18.1 Electrostatic Fields 75918.2 Use of Conformal Mapping. Modeling 76318.3 Heat Problems 76718.4 Fluid Flow 77118.5 Poisson’s Integral Formula for Potentials 77718.6 General Properties of Harmonic Functions.Uniqueness Theorem for the Dirichlet Problem 781Chapter 18 Review Questions and Problems 785Summary of Chapter 18 786P A R T E Numeric Analysis 787Software 788
CHAPTER 19 Numerics in General 790
19.1 Introduction 79019.2 Solution of Equations by Iteration 79819.3 Interpolation 80819.4 Spline Interpolation 82019.5 Numeric Integration and Differentiation 827Chapter 19 Review Questions and Problems 841Summary of Chapter 19 842
CHAPTER 20 Numeric Linear Algebra 844
20.1 Linear Systems: Gauss Elimination 84420.2 Linear Systems: LU-Factorization, Matrix Inversion 85220.3 Linear Systems: Solution by Iteration 85820.4 Linear Systems: Ill-Conditioning, Norms 86420.5 Least Squares Method 87220.6 Matrix Eigenvalue Problems: Introduction 87620.7 Inclusion of Matrix Eigenvalues 87920.8 Power Method for Eigenvalues 88520.9 Tridiagonalization and QR-Factorization 888Chapter 20 Review Questions and Problems 896Summary of Chapter 20 898
CHAPTER 21 Numerics for ODEs and PDEs 900
21.1 Methods for First-Order ODEs 90121.2 Multistep Methods 91121.3 Methods for Systems and Higher Order ODEs 915Contents xix21.4 Methods for Elliptic PDEs 92221.5 Neumann and Mixed Problems. Irregular Boundary 93121.6 Methods for Parabolic PDEs 93621.7 Method for Hyperbolic PDEs 942Chapter 21 Review Questions and Problems 945Summary of Chapter 21 946
P A R T F Optimization, Graphs 949
CHAPTER 22 Unconstrained Optimization. Linear Programming 950
22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 95122.2 Linear Programming 95422.3 Simplex Method 95822.4 Simplex Method: Difficulties 962Chapter 22 Review Questions and Problems 968Summary of Chapter 22 969
CHAPTER 23 Graphs. Combinatorial Optimization 970
23.1 Graphs and Digraphs 97023.2 Shortest Path Problems. Complexity 97523.3 Bellman’s Principle. Dijkstra’s Algorithm 98023.4 Shortest Spanning Trees: Greedy Algorithm 98423.5 Shortest Spanning Trees: Prim’s Algorithm 98823.6 Flows in Networks 99123.7 Maximum Flow: Ford–Fulkerson Algorithm 99823.8 Bipartite Graphs. Assignment Problems 1001Chapter 23 Review Questions and Problems 1006Summary of Chapter 23 1007P A R T G Probability, Statistics 1009Software 1009
CHAPTER 24 Data Analysis. Probability Theory 1011
24.1 Data Representation. Average. Spread 101124.2 Experiments, Outcomes, Events 101524.3 Probability 101824.4 Permutations and Combinations 102424.5 Random Variables. Probability Distributions 102924.6 Mean and Variance of a Distribution 103524.7 Binomial, Poisson, and Hypergeometric Distributions 103924.8 Normal Distribution 104524.9 Distributions of Several Random Variables 1051Chapter 24 Review Questions and Problems 1060Summary of Chapter 24 1060
CHAPTER 25 Mathematical Statistics 1063
25.1 Introduction. Random Sampling 106325.2 Point Estimation of Parameters 106525.3 Confidence Intervals 1068xx Contents25.4 Testing Hypotheses. Decisions 107725.5 Quality Control 108725.6 Acceptance Sampling 109225.7 Goodness of Fit. 2-Test 109625.8 Nonparametric Tests 110025.9 Regression. Fitting Straight Lines. Correlation 1103Chapter 25 Review Questions and Problems 1111Summary 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 A63A3.2 Partial Derivatives A69A3.3 Sequences and Series A72A3.4 Grad, Div, Curl, 2 in Curvilinear Coordinates A74
APPENDIX 4 Additional Proofs A77
APPENDIX 5 Tables A97
INDEX I1
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