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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. Cover Title Page Copyright Preface Contents PART A - Ordinary Differential Equations (ODEs) Chapter 1: First-Order ODEs 1.1 Basic Concepts. Modeling 1.2 Geometric Meaning of y'=f(x,y). Direction Fields, Euler’s Method 1.3 Separable ODEs. Modeling 1.4 Exact ODEs. Integrating Factors 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 1.6 Orthogonal Trajectories. 1.7 Existence and Uniqueness of Solutions for Initial Value Problems Chapter 2: Second-Order Linear ODEs 2.1 Homogeneous Linear ODEs of Second Order 2.2 Homogeneous Linear ODEs with Constant Coefficients 2.3 Differential Operators. 2.4 Modeling of Free Oscillations of a Mass–Spring System 2.5 Euler–Cauchy Equations 2.6 Existence and Uniqueness of Solutions. Wronskian 2.7 Nonhomogeneous ODEs 2.8 Modeling: Forced Oscillations. Resonance 2.9 Modeling: Electric Circuits 2.10 Solution by Variation of Parameters Chapter 3: Higher Order Linear ODEs 3.1 Homogeneous Linear ODEs 3.2 Homogeneous Linear ODEs with Constant Coefficients 3.3 Nonhomogeneous Linear ODEs Chapter 4: Systems of ODEs. Phase Plane. Qualitative Methods 4.0 For Reference: Basics of Matrices and Vectors 4.1 Systems of ODEs as Models in Engineering Applications 4.2 Basic Theory of Systems of ODEs. Wronskian 4.3 Constant-Coefficient Systems. Phase Plane Method 4.4 Criteria for Critical Points. Stability 4.5 Qualitative Methods for Nonlinear Systems 4.6 Nonhomogeneous Linear Systems of ODEs Chapter 5: Series Solutions of ODEs. Special Functions 5.1 Power Series Method 5.2 Legendre’s Equation. Legendre Polynomials Pn(X) 5.3 Extended Power Series Method: Frobenius Method 5.4 Bessel’s Equation. Bessel Functions Jv(x) 5.5 Bessel Functions Yv(x). General Solution Chapter 6: Laplace Transforms 6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) 6.2 Transforms of Derivatives and Integrals. ODEs 6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting) 6.4 Short Impulses. Dirac’s Delta Function. Partial Fractions 6.5 Convolution. Integral Equations 6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients 6.7 Systems of ODEs 6.8 Laplace Transform: General Formulas 6.9 Table of Laplace Transforms PART B - Linear Algebra. Vector Calculus Chapter 7: Linear Algebra: Matrices, Vectors, Determinants. Linear Systems 7.1 Matrices, Vectors: Addition and Scalar Multiplication 7.2 Matrix Multiplication 7.3 Linear Systems of Equations. Gauss Elimination 7.4 Linear Independence. Rank of a Matrix. Vector Space 7.5 Solutions of Linear Systems: Existence, Uniqueness 7.6 For Reference: Secondand Third-Order Determinants 7.7 Determinants. Cramer’s Rule 7.8 Inverse of a Matrix. Gauss–Jordan Elimination 7.9 Vector Spaces, Inner Product Spaces, Linear Transformations Chapter 8: Linear Algebra: Matrix Eigenvalue Problems 8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors 8.2 Some Applications of Eigenvalue Problems 8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices 8.4 Eigenbases. Diagonalization. Quadratic Forms 8.5 Complex Matrices and Forms. Chapter 9: Vector Differential Calculus. Grad, Div, Curl 9.1 Vectors in 2-Space and 3-Space 9.2 Inner Product (Dot Product) 9.3 Vector Product (Cross Product) 9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives 9.5 Curves. Arc Length. Curvature. Torsion 9.6 Calculus Review: Functions of Several Variables. 9.7 Gradient of a Scalar Field. Directional Derivative 9.8 Divergence of a Vector Field 9.9 Curl of a Vector Field Chapter 10: Vector Integral Calculus. Integral Theorems 10.1 Line Integrals 10.2 Path Independence of Line Integrals 10.3 Calculus Review: Double Integrals. 10.4 Green’s Theorem in the Plane 10.5 Surfaces for Surface Integrals 10.6 Surface Integrals 10.7 Triple Integrals. Divergence Theorem of Gauss 10.8 Further Applications of the Divergence Theorem 10.9 Stokes’s Theorem PART C - Fourier Analysis. Partial Differential Equations (PDEs) Chapter 11: Fourier Analysis 11.1 Fourier Series 11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 11.3 Forced Oscillations 11.4 Approximation by Trigonometric Polynomials 11.5 Sturm–Liouville Problems. Orthogonal Functions 11.6 Orthogonal Series. Generalized Fourier Series 11.7 Fourier Integral 11.8 Fourier Cosine and Sine Transforms 11.9 Fourier Transform. Discrete and Fast Fourier Transforms 11.10 Tables of Transforms Chapter 12: Partial Differential Equations (PDEs) 12.1 Basic Concepts of PDEs 12.2 Modeling: Vibrating String, Wave Equation 12.3 Solution by Separating Variables. Use of Fourier Series 12.4 D’Alembert’s Solution of the Wave Equation. Characteristics 12.5 Modeling: Heat Flow from a Body in Space. Heat Equation 12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem 12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms 12.8 Modeling: Membrane, Two-Dimensional Wave Equation 12.9 Rectangular Membrane. Double Fourier Series 12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series 12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential Solution of PDEs by Laplace Transforms PART D - Complex Analysis Chapter 13: Complex Numbers and Functions. Complex Differentiation 13.1 Complex Numbers and Their Geometric Representation 13.2 Polar Form of Complex Numbers. Powers and Roots 13.3 Derivative. Analytic Function 13.4 Cauchy–Riemann Equations. Laplace’s Equation 13.5 Exponential Function 13.6 Trigonometric and Hyperbolic Functions. Euler’s Formula 13.7 Logarithm. General Power. Principal Value Chapter 14: Complex Integration 14.1 Line Integral in the Complex Plane 14.2 Cauchy’s Integral Theorem 14.3 Cauchy’s Integral Formula 14.4 Derivatives of Analytic Functions Chapter 15: Power Series, Taylor Series 15.1 Sequences, Series, Convergence Tests 15.2 Power Series 15.3 Functions Given by Power Series 15.4 Taylor and Maclaurin Series 15.5 Uniform Convergence. Chapter 16: Laurent Series. Residue Integration 16.1 Laurent Series 16.2 Singularities and Zeros. Infinity 16.3 Residue Integration Method 16.4 Residue Integration of Real Integrals Chapter 17: Conformal Mapping 17.1 Geometry of Analytic Functions: Conformal Mapping 17.2 Linear Fractional Transformations (Möbius Transformations) 17.3 Special Linear Fractional Transformations 17.4 Conformal Mapping by Other Functions 17.5 Riemann Surfaces. Chapter 18: Complex Analysis and Potential Theory 18.1 Electrostatic Fields 18.2 Use of Conformal Mapping. Modeling 18.3 Heat Problems 18.4 Fluid Flow 18.5 Poisson’s Integral Formula for Potentials 18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem PART E - Numeric Analysis Software Chapter 19: Numerics in General 19.1 Introduction 19.2 Solution of Equations by Iteration 19.3 Interpolation 19.4 Spline Interpolation 19.5 Numeric Integration and Differentiation Chapter 20: Numeric Linear Algebra 20.1 Linear Systems: Gauss Elimination 20.2 Linear Systems: LU-Factorization, Matrix Inversion 20.3 Linear Systems: Solution by Iteration 20.4 Linear Systems: Ill-Conditioning, Norms 20.5 Least Squares Method 20.6 Matrix Eigenvalue Problems: Introduction 20.7 Inclusion of Matrix Eigenvalues 20.8 Power Method for Eigenvalues 20.9 Tridiagonalization and QR-Factorization Chapter 21: Numerics for ODEs and PDEs 21.1 Methods for First-Order ODEs 21.2 Multistep Methods 21.3 Methods for Systems and Higher Order ODEs 21.4 Methods for Elliptic PDEs 21.5 Neumann and Mixed Problems. Irregular Boundary 21.6 Methods for Parabolic PDEs 21.7 Method for Hyperbolic PDEs PART F - Optimization, Graphs Chapter 22: Unconstrained Optimization. Linear Programming 22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 22.2 Linear Programming 22.3 Simplex Method 22.4 Simplex Method: Difficulties Chapter 23: Graphs. Combinatorial Optimization 23.1 Graphs and Digraphs 23.2 Shortest Path Problems. Complexity 23.3 Bellman’s Principle. Dijkstra’s Algorithm 23.4 Shortest Spanning Trees: Greedy Algorithm 23.5 Shortest Spanning Trees: Prim’s Algorithm 23.6 Flows in Networks 23.7 Maximum Flow: Ford–Fulkerson Algorithm 23.8 Bipartite Graphs. Assignment Problems PART G - Probability, Statistics Additional Software for Probability and Statistics Chapter 24: Data Analysis. Probability Theory 24.1 Data Representation. Average. Spread 24.2 Experiments, Outcomes, Events 24.3 Probability 24.4 Permutations and Combinations 24.5 Random Variables. Probability Distributions 24.6 Mean and Variance of a Distribution 24.7 Binomial, Poisson, and Hypergeometric Distributions 24.8 Normal Distribution 24.9 Distributions of Several Random Variables Chapter 25: Mathematical Statistics 25.1 Introduction. Random Sampling 25.2 Point Estimation of Parameters 25.3 Confidence Intervals 25.4 Testing of Hypotheses. Decisions 25.5 Quality Control 25.6 Acceptance Sampling 25.7 Goodness of Fit. -Test 25.8 Nonparametric Tests 25.9 Regression. Fitting Straight Lines. Correlation Appendix 1: References Appendix 2: Answers to Odd-Numbered Problems Appendix 3: Auxiliary Material Appendix 4: Additional Proofs Appendix 5: Tables Index Photo Credits 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.