Advanced Engineering Mathematics, International Adaptation

ERWIN KREYSZIG, Professor of Mathematics, Ohio State University, Columbus, Ohio.

• TABLE OF CONTENTS PART A Ordinary Differential Equations (ODEs)  CHAPTER 1 First-Order ODEs1.1 Basic Concepts. Modeling1.2 Geometric Meaning of y’ = ƒ(x, y). Direction Fields, Euler’s Method1.3 Separable ODEs. Modeling1.4 Exact ODEs. Integrating Factors1.5 Linear ODEs. Bernoulli Equation. Population Dynamics1.6 Orthogonal Trajectories. Optional1.7 Existence and Uniqueness of Solutions for Initial Value ProblemsSustainability and Ethical ConsiderationsChapter 1 Review Questions and ProblemsSummary of Chapter 1  CHAPTER 2 Second-Order Linear ODEs2.1 Homogeneous Linear ODEs of Second Order2.2 Homogeneous Linear ODEs with Constant Coefficients2.3 Differential Operators. Optional2.4 Modeling of Free Oscillations of a Mass–Spring System2.5 Euler–Cauchy Equations2.6 Existence and Uniqueness of Solutions. Wronskian2.7 Nonhomogeneous ODEs2.8 Modeling: Forced Oscillations. Resonance2.9 Modeling: Electric Circuits2.10 Solution by Variation of ParametersSustainability and Ethical ConsiderationsChapter 2 Review Questions and ProblemsSummary of Chapter 2  CHAPTER 3 Higher Order Linear ODEs3.1 Homogeneous Linear ODEs3.2 Homogeneous Linear ODEs with Constant Coefficients3.3 Nonhomogeneous Linear ODEsSustainability and Ethical ConsiderationsChapter 3 Review Questions and ProblemsSummary of Chapter 3  CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods4.0 For Reference: Basics of Matrices and Vectors4.1 Systems of ODEs as Models in Engineering Applications4.2 Basic Theory of Systems of ODEs. Wronskian4.3 Constant-Coefficient Systems. Phase Plane Method4.4 Criteria for Critical Points. Stability4.5 Qualitative Methods for Nonlinear Systems4.6 Nonhomogeneous Linear Systems of ODEsSustainability and Ethical ConsiderationsChapter 4 Review Questions and ProblemsSummary of Chapter 4  CHAPTER 5 Series Solutions of ODEs. Special Functions5.1 Power Series Method5.2 Legendre’s Equation. Legendre Polynomials (x)5.3 Extended Power Series Method: Frobenius Method5.4 Bessel’s Equation. Bessel Functions (x)5.5 Bessel Functions of the (x). General SolutionSustainability and Ethical ConsiderationsChapter 5 Review Questions and ProblemsSummary of Chapter 5  CHAPTER 6 Laplace Transforms6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting)6.2 Transforms of Derivatives and Integrals. ODEs6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting)6.4 Short Impulses. Dirac’s Delta Function. Partial Fractions6.5 Convolution. Integral Equations6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients6.7 Systems of ODEs6.8 Laplace Transform: General Formulas6.9 Table of Laplace TransformsSustainability and Ethical ConsiderationsChapter 6 Review Questions and ProblemsSummary of Chapter 6  PART B Linear Algebra. Vector Calculus  CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems7.1 Matrices, Vectors: Addition and Scalar Multiplication7.2 Matrix Multiplication7.3 Linear Systems of Equations. Gauss Elimination7.4 Linear Independence. Rank of a Matrix. Vector Space7.5 Solutions of Linear Systems: Existence, Uniqueness7.6 For Reference: Second- and Third-Order Determinants7.7 Determinants. Cramer’s Rule7.8 Inverse of a Matrix. Gauss–Jordan Elimination7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. OptionalSustainability and Ethical ConsiderationsChapter 7 Review Questions and ProblemsSummary of Chapter 7  CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors8.2 Some Applications of Eigenvalue Problems8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices8.4 Eigenbases. Diagonalization. Quadratic Forms8.5 Complex Matrices and Forms. OptionalSustainability and Ethical ConsiderationsChapter 8 Review Questions and ProblemsSummary of Chapter 8  CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl9.1 Vectors in 2-Space and 3-Space9.2 Inner Product (Dot Product)9.3 Vector Product (Cross Product)9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives9.5 Curves. Arc Length. Curvature. Torsion9.6 Calculus Review: Functions of Several Variables. Optional9.7 Gradient of a Scalar Field. Directional Derivative9.8 Divergence of a Vector Field9.9 Curl of a Vector FieldSustainability and Ethical ConsiderationsChapter 9 Review Questions and ProblemsSummary of Chapter 9  CHAPTER 10 Vector Integral Calculus. Integral Theorems10.1 Line Integrals10.2 Path Independence of Line Integrals10.3 Calculus Review: Double Integrals. Optional10.4 Green’s Theorem in the Plane10.5 Surfaces for Surface Integrals10.6 Surface Integrals10.7 Triple Integrals. Divergence Theorem of Gauss10.8 Further Applications of the Divergence Theorem10.9 Stokes’s TheoremSustainability and Ethical ConsiderationsChapter 10 Review Questions and ProblemsSummary of Chapter 10  PART C Fourier Analysis. Partial Differential Equations (PDEs)  CHAPTER 11 Fourier Analysis11.1 Fourier Series11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions11.3 Forced Oscillations11.4 Approximation by Trigonometric Polynomials11.5 Sturm–Liouville Problems. Orthogonal Functions11.6 Orthogonal Series. Generalized Fourier Series11.7 Fourier Integral11.8 Fourier Cosine and Sine Transforms11.9 Fourier Transform. Discrete and Fast Fourier Transforms11.10 Tables of TransformsSustainability and Ethical ConsiderationsChapter 11 Review Questions and ProblemsSummary of Chapter 11  CHAPTER 12 Partial Differential Equations (PDEs)12.1 Basic Concepts of PDEs12.2 Modeling: Vibrating String, Wave Equation12.3 Solution by Separating Variables. Use of Fourier Series12.4 D’Alembert’s Solution of the Wave Equation. Characteristics12.5 Modeling: Heat Flow from a Body in Space. Heat Equation12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms12.8 Modeling: Membrane, Two-Dimensional Wave Equation12.9 Rectangular Membrane. Double Fourier Series12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential12.12 Solution of PDEs by Laplace TransformsSustainability and Ethical ConsiderationsChapter 12 Review Questions and ProblemsSummary of Chapter 12  PART D Complex Analysis  CHAPTER 13 Complex Numbers and Functions. Complex Differentiation13.1 Complex Numbers and Their Geometric Representation13.2 Polar Form of Complex Numbers. Powers and Roots13.3 Derivative. Analytic Function13.4 Cauchy–Riemann Equations. Laplace’s Equation13.5 Exponential Function13.6 Trigonometric and Hyperbolic Functions. Euler’s Formula13.7 Logarithm. General Power. Principal ValueSustainability and Ethical ConsiderationsChapter 13 Review Questions and ProblemsSummary of Chapter 13  CHAPTER 14 Complex Integration14.1 Line Integral in the Complex Plane14.2 Cauchy’s Integral Theorem14.3 Cauchy’s Integral Formula14.4 Derivatives of Analytic FunctionsSustainability and Ethical ConsiderationsChapter 14 Review Questions and ProblemsSummary of Chapter 14  CHAPTER 15 Power Series, Taylor Series15.1 Sequences, Series, Convergence Tests15.2 Power Series15.3 Functions Given by Power Series15.4 Taylor and Maclaurin Series15.5 Uniform Convergence. OptionalSustainability and Ethical ConsiderationsChapter 15 Review Questions and ProblemsSummary of Chapter 15  CHAPTER 16 Laurent Series. Residue Integration16.1 Laurent Series16.2 Singularities and Zeros. Infinity16.3 Residue Integration Method16.4 Residue Integration of Real IntegralsSustainability and Ethical ConsiderationsChapter 16 Review Questions and ProblemsSummary of Chapter 16  CHAPTER 17 Conformal Mapping17.1 Geometry of Analytic Functions: Conformal Mapping17.2 Linear Fractional Transformations (Möbius Transformations)17.3 Special Linear Fractional Transformations17.4 Conformal Mapping by Other Functions17.5 Riemann Surfaces. OptionalSustainability and Ethical ConsiderationsChapter 17 Review Questions and ProblemsSummary of Chapter 17  CHAPTER 18 Complex Analysis and Potential Theory18.1 Electrostatic Fields18.2 Use of Conformal Mapping. Modeling18.3 Heat Problems18.4 Fluid Flow18.5 Poisson’s Integral Formula for Potentials18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet ProblemSustainability and Ethical ConsiderationsChapter 18 Review Questions and ProblemsSummary of Chapter 18  PART E Numeric Analysis Software  CHAPTER 19 Numerics in General19.1 Introduction19.2 Solution of Equations by Iteration19.3 Interpolation19.4 Spline Interpolation19.5 Numeric Integration and DifferentiationSustainability and Ethical ConsiderationsChapter 19 Review Questions and ProblemsSummary of Chapter 19  CHAPTER 20 Numeric Linear Algebra20.1 Linear Systems: Gauss Elimination20.2 Linear Systems: LU-Factorization, Matrix Inversion20.3 Linear Systems: Solution by Iteration20.4 Linear Systems: Ill-Conditioning, Norms20.5 Least Squares Method20.6 Matrix Eigenvalue Problems: Introduction20.7 Inclusion of Matrix Eigenvalues20.8 Power Method for Eigenvalues20.9 Tridiagonalization and QR-FactorizationSustainability and Ethical ConsiderationsChapter 20 Review Questions and ProblemsSummary of Chapter 20  CHAPTER 21 Numerics for ODEs and PDEs21.1 Methods for First-Order ODEs21.2 Multistep Methods21.3 Methods for Systems and Higher Order ODEs21.4 Methods for Elliptic PDEs21.5 Neumann and Mixed Problems. Irregular Boundary21.6 Methods for Parabolic PDEs21.7 Method for Hyperbolic PDEsSustainability and Ethical ConsiderationsChapter 21 Review Questions and ProblemsSummary of Chapter 21  PART F Optimization, Graphs  CHAPTER 22 Unconstrained Optimization. Linear Programming22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent22.2 Linear Programming22.3 Simplex Method22.4 Simplex Method: DifficultiesSustainability and Ethical ConsiderationsChapter 22 Review Questions and ProblemsSummary of Chapter 22  CHAPTER 23 Graphs. Combinatorial Optimization23.1 Graphs and Digraphs23.2 Shortest Path Problems. Complexity23.3 Bellman’s Principle. Dijkstra’s Algorithm23.4 Shortest Spanning Trees: Greedy Algorithm23.5 Shortest Spanning Trees: Prim’s Algorithm23.6 Flows in Networks23.7 Maximum Flow: Ford–Fulkerson Algorithm23.8 Bipartite Graphs. Assignment ProblemsSustainability and Ethical ConsiderationsChapter 23 Review Questions and ProblemsSummary of Chapter 23  PART G Probability, Statistics              (available online)Software  CHAPTER 24 Data Analysis. Probability Theory24.1 Data Representation. Average. Spread24.2 Experiments, Outcomes, Events24.3 Probability24.4 Permutations and Combinations24.5 Random Variables. Probability Distributions24.6 Mean and Variance of a Distribution24.7 Binomial, Poisson, and Hypergeometric Distributions24.8 Normal Distribution24.9 Distributions of Several Random VariablesSustainability and Ethical ConsiderationsChapter 24 Review Questions and ProblemsSummary of Chapter 24  CHAPTER 25 Mathematical Statistics25.1 Introduction. Random Sampling25.2 Point Estimation of Parameters25.3 Confidence Intervals25.4 Testing Hypotheses. Decisions25.5 Quality Control25.6 Acceptance Sampling25.7 Goodness of Fit. χ2-Test25.8 Nonparametric Tests25.9 Regression. Fitting Straight Lines. CorrelationSustainability and Ethical ConsiderationsChapter 25 Review Questions and ProblemsSummary of Chapter 25  PROJECTs  APPENDIX 1 References  APPENDIX 2 Answers to Odd-Numbered Problems (available online)  APPENDIX 3 Auxiliary MaterialA3.1 Formulas for Special FunctionsA3.2 Partial DerivativesA3.3 Sequences and SeriesA3.4 Grad, Div, Curl,  in Curvilinear Coordinates  APPENDIX 4 Additional Proofs  APPENDIX 5 Tables  APPENDIX 6 Emerging Topics in Applied Mathematics  INDEX  PHOTO CREDITS