Advanced Engineering Mathematics, by Robert Lopez

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Contents
Preface
UNIT I. Ordinary Differential Equations-Part One
Chapter 1 First-Order Differential Equations: 1.1 Introduction; 1.2 Terminology; 1.3 The Direction Field; 1.4 Picard Iteration; 1.5 Existence and Uniqueness for the Initial Value Problem
Chapter 2 Models Containing ODEs: 2.1 Exponential Growth and Decay; 2.2 Logistic Models; 2.3 Mixing Tank Problems-Constant and Variable Volumes; 2.4 Newton's Law of Cooling
Chapter 3 Methods for Solving First-Order ODEs: 3.1 Separation of Variables; 3.2 Equations with Homogeneous Coefficients; 3.3 Exact Equations; 3.4 Integrating Factors and the First-Order Equations; 3.5 Variation of Parameters and the First-Order Linear Equation; 3.6 The Bernoulli Equation
Chapter 4 Numeric Methods for Solving First-Order ODEs: 4.1 Fixed-Step methods-Order and Error; 4.2 The Euler Method; 4.3 Taylor Series Methods; 4.4 Runge-Kutta Methods; 4.5 Adams-Bashforth Multistep Methods; 4.6 Adams-Moulton Predictor-Corrector Methods; 4.7 Milne's Method; 4.8 rkf45, the Runge-Kutta-Fehlberg Method
Chapter 5 Second-Order Differential Equations: 5.1 Springs 'n' Things; 5.2 The Initial Value Problem; 5.3 Overview of Solution Process; 5.4 Linear Dependence and Independence; 5.5 Free Undamped Motion; 5.6 Free Damped Motion; 5.7 Reduction of Order and Higher-Order Equations; 5.8 The Bobbing Cylinder; 5.9 Forced Motion and Variation of Parameters; 5.10 Forced Motion and Undetermined Coefficients; 5.11 Resonance; 5.12 The Euler Equation; 5.13 The Green's Function Technique for IVPs
Chapter 6 The Laplace Transform: 6.1 Definition and Examples; 6.2 Transform of Derivatives; 6.3 First Shifting Law; 6.4 Operational Laws; 6.5 Heaviside Functions and the Second Shifting Law; 6.6 Pulses and the Third Shifting Law; 6.7 Transforms of Periodic Functions; 6.8 Convolution and the Convolution Theorem; 6.9 Convolution Products by the Convolution Theorem; 6.10 The Dirac Delta Function; 6.11 Transfer Function, Fundamental Solution, and the Green's Function; Go to Top
UNIT II. Infinite Series
Chapter 7 Sequences and Series of Numbers: 7.1 Sequences; 7.2 Infinite Series; 7.3 Series with Positive Terms; 7.4 Series with both Negative and Positive Terms
Chapter 8 Sequences and Series of Functions: 8.1 Sequences of Functions; 8.2 Pointwise Convergence; 8.3 Uniform Convergence; 8.4 Convergence in the Mean; 8.5 Series of Functions
Chapter 9 Power Series: 9.1 Taylor Polynomials; 9.2 Taylor Series; 9.3 Termwise Operations on Taylor Series
Chapter 10 Fourier Series: 10.1 General Formalism; 10.2 Termwise Integration and Differentiation; 10.3 Odd and Even Functions and their Fourier Series; 10.4 Sine Series and Cosine Series; 10.5 Periodically Driven Damped Oscillator; 10.6 Optimizing Property of Fourier Series; 10.7 Fourier-Legendre Series
Chapter 11 Asymptotic Series: 11.1 Computing with Divergent Series; 11.2 Definitions; 11.3 Operations with Asymptotic Series; Go to Top
UNIT III. Ordinary Differential Equations-Part Two
Chapter 12 Systems of First-Order ODEs: 12.1 Mixing tanks-Closed Systems; 12.2 Mixing tanks-Open Systems; 12.3 Vector Structure of Solutions; 12.4 Determinants and Cramer's Rule; 12.5 Solving Linear Algebraic Equations; 12.6 Homogeneous Equations and the Null Space; 12.7 Inverses; 12.8 Vectors and the Laplace Transform; 12.9 The Matrix Exponential; 12.10 Eigenvalues and Eigenvectors; 12.11 Solutions by Eigenvalues and Eigenvectors; 12.12 Finding Eigenvalues and Eigenvectors; 12.13 System versus. Second-Order ODE; 12.14 Complex Eigenvalues; 12.15 The Deficient Case; 12.16 Diagonalization and Uncoupling; 12.17 A Coupled Linear Oscillator; 12.18 Nonhomogeneous Systems and Variation of Parameters; 12.19 Phase Portraits; 12.20 Stability; 12.21 Nonlinear Systems; 12.22 Linearization; 12.23 The Nonlinear Pendulum
Chapter 13 Numerical Techniques: First-Order Systems and Second-Order ODEs: 13.1 Runge-Kutta-Nystrom; 13.2 rk4 for First-Order Systems
Chapter 14 Series Solutions: 14.1 Power series; 14.2 Asymptotic solutions; 14.3 Perturbation Solution of an Algebraic Equation; 14.4 Poincaré Perturbation Solution for Differential Equations; 14.5 The Nonlinear Spring and Lindstedt's Method; 14.6 The Method of Krylov and Bogoliubov
Chapter 15 Boundary Value Problems: 15.1 Analytic Solutions; 15.2 Numeric Solutions; 15.3 Least-squares, Rayleigh-Ritz, Galerkin, and Collocation Techniques; 15.4 Finite Elements
Chapter 16 The Eigenvalue Problem: 16.1 Regular Sturm-Liouville Problems; 16.2 Bessel's Equation; 16.3 Legendre's Equation; 16.4 Solution by Finite Differences; Go to Top
UNIT IV. Vector Calculus
Chapter 17 Space Curves: 17.1 Curves and Their Tangent Vectors; 17.2 Arc Length; 17.3 Curvature; 17.4 Principal Normal and Binormal Vectors; 17.5 Resolution of R" into Tangential and Normal Components; 17.6 Applications to Dynamics
Chapter 18 The Gradient Vector: 18.1 Visualizing Vector Fields and Their Flows; 18.2 The Directional Derivative and Gradient Vector; 18.3 Properties of the Gradient Vector; 18.4 Lagrange Multipliers; 18.5 Conservative Forces and the Scalar Potential
Chapter 19 Line Integrals in the Plane: 19.1 Work and Circulation; 19.2 Flux through a Plane Curve
Chapter 20 Additional Vector Differential Operators: 20.1 Divergence and Its Meaning; 20.2 Curl and Its Meaning; 20.3 Products-One Del and Two Operands; 20.4 Products-Two Dels and One Operand
Chapter 21 Integration: 21.1 Surface Area; 21.2 Surface Integrals and Surface Flux; 21.3 The Divergence Theorem and the Theorems of Green and Stokes; 21.4 Green's Theorem; 21.5 Conservative, Solenoidal, and Irrotational Fields; 21.6 Integral Equivalents of div, grad, and curl
Chapter 22 Non-Cartesian Coordinates: 22.1 Mappings and Changes of Coordinates; 22.2 Vector Operators in Polar Coordinates; 22.3 Vector Operators in Cylindrical and Spherical Csoordinates
Chapter 23 Miscellaneous Results: 23.1 Gauss' Theorem; 23.2 Surface Area for Parametrically Given Surfaces; 23.3 The Equation of Continuity; 23.4 Green's Identities; Go to Top
UNIT V. Boundary Value Problems for PDEs
Chapter 24 Wave Equation: 24.1 The Plucked String; 24.2 The Struck String; 24.3 D'Alembert's Solution; 24.4 Derivation of the Wave Equation; 24.5 Longitudinal Vibrations in an Elastic Rod; 24.6 Finite-Difference Solution of the One-Dimensional Wave Equation
Chapter 25 Heat Equation: 25.1 One-Dimensional Heat Diffusion; 25.2 Derivation of the One-Dimensional Heat Equation; 25.3 Heat Flow in a Rod with Insulated Ends; 25.4 Finite-Difference Solution of the One-Dimensional Heat Equation
Chapter 26 Laplace's Equation on a Rectangle: 26.1 Nonzero Temperature on the Bottom Edge; 26.2 Nonzero Temperature on the Top Edge; 26.3 Nonzero Temperature on the Left Edge; 26.4 Finite-Difference Solution of Laplace's Equation on a Rectangle
Chapter 27 Nonhomogeneous Boundary Value Problems: 27.1 One-Dimensional Heat Equation with Different Endpoint Temperatures; 27.2 One-Dimensional Heat Equation with Time-Varying Endpoint Temperatures
Chapter 28 Time-Dependent Problems in Two Spatial Dimensions: 28.1 Oscillations of a Rectangular Membrane; 28.2 Time-Varying Temperatures on a Rectangular Plate
Chapter 29 Separation of Variables in Non-Cartesian Coordinates: 29.1 Laplace's Equation in a Disk; 29.2 Laplace's Equation in a Cylinder; 29.3 The Circular Drumhead; 29.4 Laplace's Equation in a Sphere; 29.5 The Spherical Dielectric
Chapter 30 Transform Techniques: 30.1 Solution by Laplace Transform; 30.2 The Fourier Integral Theorem; 30.3 The Fourier Transform; 30.4 Wave Equation on the Infinite String-Solution by Fourier Transform; 30.5 Heat Equation on the Infinite Rod-Solution by Fourier Transform; 30.6 Laplace's Equation on the Infinite Strip-Solution by Fourier Transform; 30.7 The Fourier Sine Transform; 30.8 The Fourier Cosine Transform; Go to Top
UNIT VI. Matrix Algebra
Chapter 31 Vectors as Arrows: 31.1 The Algebra and Geometry of Vectors; 31.2 Inner and Dot Products; 31.3 The Cross-Product
Chapter 32 Change of Coordinates: 32.1 Change of Basis; 32.2 Rotations and Orthogonal Matrices; 32.3 Change of Coordinates; 32.4 Reciprocal Bases and Gradient Vectors; 32.5 Gradient Vectors and the Covariant Transformation Law
Chapter 33 Matrix Computations: 33.1 Summary; 33.2 Projections; 33.3 The Gram-Schmidt Orthogonalization Process; 33.4 Quadratic Forms; 33.5 Vector and Matrix Norms; 33.6 Least Squares
Chapter 34 Matrix Factorizations: 34.1 LU Decomposition; 34.2 PJP^-1 and Jordan Canonical Form; 34.3 QR Decomposition; 34.4 QR Algorithm for Finding Eigenvalues; 34.5 SVD, The Singular Value Decomposition; 34.6 Minimum-Length Least-Squares Solution, and the Pseudoinverse; Go to Top
UNIT VII. Complex Variables
Chapter 35 Fundamentals: 35.1 Complex Numbers; 35.2 The Function w = f(z) = z²; 35.3 The Function w = f(z) = z³; 35.4 The Exponential Function; 35.5 The Complex Logarithm; 35.6 Complex Exponents; 35.7 Trigonometric and Hyperbolic Functions; 35.8 Inverses of Trigonometric and Hyperbolic Functions; 35.9 Differentiation and the Cauchy-Riemann Equations; 35.10 Analytic and Harmonic Functions; 35.11 Integration; 35.12 Series in Powers of z; 35.13 The Calculus of Residues
Chapter 36 Applications: 36.1 Evaluation of Integrals; 36.2 The Laplace Transform; 36.3 Fourier Series and the Fourier Transform; 36.4 The Root Locus; 36.5 The Nyquist Stability Criterion; 36.6 Conformal Mapping; 36.7 The Joukowski Map; 36.8 Solving the Dirichlet Problem by Conformal Mapping; 36.9 Planar Fluid Flow; 36.10 Conformal Mapping of Elementary Flows; Go to Top
UNIT VIII. Numerical Methods
Chapter 37 Equations in One Variable-Preliminaries: 37.1 Accuracy and Errors; 37.2 Rate of Convergence
Chapter 38 Equations in One Variable-Methods: 38.1 Fixed-Point Iteration; 38.2 The Bisection Method; 38.3 Newton-Raphson Iteration; 38.4 The Secant Method; 38.5 Muller's Method
Chapter 39 Systems of Equations: 39.1 Gaussian Arithmetic; 39.2 Condition Numbers; 39.3 Iterative Improvement; 39.4 The Method of Jacobi; 39.5 Gauss-Seidel Iteration; 39.6 Relaxation and SOR; 39.7 Iterative Mmethods for Nonlinear Systems; 39.8 Newton's Iteration for Nonlinear Systems
Chapter 40 Interpolation: 40.1 Lagrange Interpolation; 40.2 Divided Differences; 40.3 Chebyshev Interpolation; 40.4 Spline Interpolation; 40.5 Bezier Curves
Chapter 41 Approximation of Continuous Functions: 41.1 Least-Squares Approximation; 41.2 Padé Approximations; 41.3 Chebyshev Approximation; 41.4 Chebyshev-Padé and Minimax Approximations
Chapter 42 Numeric Differentiation: 42.1 Basic Formulas; 42.2 Richardson Extrapolation
Chapter 43 Numeric Integration: 43.1 Methods from Elementary Calculus; 43.2 Recursive Trapezoid rule and Romberg Integration; 43.3 Gauss-Legendre Quadrature; 43.4 Adaptive Quadrature; 43.5 Iterated Integrals
Chapter 44 Approximation of Discrete Data: 44.1 Least-Squares Regression Line; 44.2 The General Linear Model; 44.3 The Role of Orthogonality; 44.4 Nonlinear Least Squares
Chapter 45 Numerical Calculation of Eigenvalues: 45.1 Power Methods; 45.2 Householder Reflections; 45.3 QR Decomposition via Householder Reflections; 45.4 Upper-Hessenberg Form, Givens Rotations, and the Shifted QR-Algorithm; 45.5 The Generalized Eigenvalue Problem; Go to Top
UNIT IX. Calculus of Variations
Chapter 46 Basic Formalisms: 46.1 Motivational Examples; 46.2 Direct Methods; 46.3 The Euler-Lagrange Equation; 46.4 First Integrals; 46.5 Derivation of the Euler-Lagrange Equation; 46.6 Transversality Conditions; 46.7 Derivation of the Transversality Conditions; 46.8 Three generalizations
Chapter 47 Constrained Optimization: 47.1 Applications of Lagrange Multipliers; 47.2 Queen Dido's Problem; 47.3 Isoperimetric Problems; 47.4 The Hanging Chain; 47.5 A Variable-Endpoint Problem; 47.6 Differential Constraints
Chapter 48 Variational Mechanics: 48.1 Hamilton's Principle; 48.2 The Simple Pendulum; 48.3 A Compound Pendulum; 48.4 The Spherical Pendulum; 48.5 Pendulum with Oscillating Support; 48.6 Legendre and Extended Legendre Transformations; 48.7 Hamilton's Canonical Equations
Answers to Selected Exercises
Bibliography
Index