Advanced engineering mathematics / Kreyszig, Erwin ; in collaboration with Herbert Kreyszig, Edward J. Norminton.

Includes bibliographical references and index.

Machine generated contents note: Ordinary Differential Equations (ODEs) -- 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. Optional -- 1.7.Existence and Uniqueness of Solutions for Initial Value Problems -- ch. 1. Review Questions and Problems -- Summary of Chapter 1 -- ch. 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. Optional -- 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 -- ch. 2. Review Questions and Problems -- Summary of Chapter 2 -- ch. 3. Higher Order Linear ODEs -- 3.1. Homogeneous Linear ODEs -- 3.2. Homogeneous Linear ODEs with Constant Coefficients -- 3.3. Nonhomogeneous Linear ODEs -- ch. 3. Review Questions and Problems -- Summary of Chapter 3 -- ch. 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 -- ch. 4. Review Questions and Problems -- ^ Summary of Chapter 4 -- ch. 5 Series Solutions of ODEs. Special Functions -- $g 5.1. $t Power Series Method -- $g 5.2. $t 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 of the Yv(x). General Solution -- ch. 5. Review Questions and Problems -- Summary of Chapter 5 -- ch. 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 -- ch. 6^ Review Questions and Problems -- Summary of Chapter 6 -- pt. B. Linear Algebra. Vector Calculus -- ch. 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: Second- and 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. Optional -- ch. 7 Review Questions and Problems -- Summary of Chapter 7 -- ch. 8. Linear Algebra: Matrix Eigenvalue Problems -- 8.1. 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. Optional -- ch. 8 -- Review Questions and Problems -- Summary of Chapter 8 -- ch. 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. Are Length. Curvature. Torsion -- 9.6. Calculus Review: Functions of Several Variables. Optional -- 9.7. Gradient of a Scalar Field. Directional Derivative -- 9.8. Divergence of a Vector Field -- 9.9. Curl of a Vector Field -- ch. 9. Review Questions and Problems -- Summary of Chapter 9 -- 10. Vector Integral Calculus. Integral Theorems -- 10.1. Line Integrals -- 10.2. Path Independence of Line Integrals -- 10.3. Calculus Review: Double Integrals. Optional -- 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 -- ch. 10. Review Questions and Problems -- Summary of Chapter 10 -- pt. C -- Fourier Analysis. Partial Differential Equations (PDEs) -- ch. 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 -- ch. 11. Review Questions and Problems -- Summary of Chapter 11 -- ch. 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. $t Laplace's Equation in Cylindrical and Spherical Coordinates. Potential -- 12.12. Solution of PDEs by Laplace Transforms -- ch. 12. Review Questions and Problems -- ^ Summary of Chapter 12 -- pt. D Complex Analysis -- ch. 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 -- ch. 13. Review Questions and Problems -- Summary of Chapter 13. ch. 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 -- ch. 14 -- Review Questions and Problems -- Summary of Chapter 14 -- ch. 15 --Power Series, Taylor Series -- 15.1. Sequences, Series, Convergence Tests -- 15.2. Power Series -- 15.3. Functions Given by Power Series -- 20.8. Power Method for Eigenvalues -- 20.9. Tridiagonalization and QR-Factorization -- ch. 20. Review Questions and Problems -- Summary of Chapter 20 -- ch. 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 -- ch. 21. Review Questions and Problems -- Summary of Chapter 21 -- pt. F. Optimization, Graphs -- ch. 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 -- ch. 22. Review Questions and Problems -- Summary of Chapter 22 -- ^ ch. 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 -- ch. 23. Review Questions and Problems -- Summary of Chapter 23 -- pt. G. Probability, Statistics -- Software -- ch. 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 -- ch. 24. Review Questions and Problems -- Summary of Chapter 24 -- ch. 25. Mathematical Statistics -- 25.1. Introduction. Random Sampling -- 25.2. Point Estimation of Parameters -- 25.3. Confidence Intervals -- 25.4. Testing Hypotheses. Decisions -- 25.5. Quality Control -- 25.6. Acceptance Sampling -- 25.7. Goodness of Fit. 2-Test -- 25.8. Nonparametric Tests -- 25.9. Regression. Fitting Straight Lines. Correlation -- ch. 25. Review Questions and Problems -- Summary of Chapter 25 -- APPENDIX 1. References -- APPENDIX 2. Answers to Odd-Numbered Problems -- APPENDIX 3. Auxiliary Material -- A3.1. Formulas for Special Functions -- A3.2. Partial Derivatives -- A3.3. Sequences and Series -- A3.4. Grad, Div, Curl, 2 in Curvilinear Coordinates -- APPENDIX 4. Additional Proofs -- APPENDIX 5. Tables.

Aimed at the junior level courses in maths and engineering departments, this edition of the well known text covers many areas such as differential equations, linear algebra, complex analysis, numerical methods, probability, and more.