Advanced Engineering Mathematics, 11ed An Indian Adaptation

PART A Ordinary Differential Equations (ODEs)

CHAPTER 1 First-Order ODEs

• Basic Concepts. Modeling
• Geometric Meaning of y’ = ƒ(x, y). Direction Fields, Euler’s Method
• Separable ODEs. Modeling
• Exact ODEs. Integrating Factors
• Linear ODEs. Bernoulli Equation. Population Dynamics
• Orthogonal Trajectories. Optional
• Existence and Uniqueness of Solutions for Initial Value Problems
• Sustainability and Ethical Considerations
• Chapter 1 Review Questions and Problems
• Summary of Chapter 1

CHAPTER 2 Second-Order Linear ODEs

• Homogeneous Linear ODEs of Second Order
• Homogeneous Linear ODEs with Constant Coefficients
• Differential Operators. Optional
• Modeling of Free Oscillations of a Mass–Spring System
• Euler–Cauchy Equations
• Existence and Uniqueness of Solutions. Wronskian
• Nonhomogeneous ODEs
• Modeling: Forced Oscillations. Resonance
• Modeling: Electric Circuits
• Solution by Variation of Parameters
• Sustainability and Ethical Considerations
• Chapter 2 Review Questions and Problems
• Summary of Chapter 2

CHAPTER 3 Higher Order Linear ODEs

• Homogeneous Linear ODEs
• Homogeneous Linear ODEs with Constant Coefficients
• Nonhomogeneous Linear ODEs
• Sustainability and Ethical Considerations
• Chapter 3 Review Questions and Problems
• Summary of Chapter 3

CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods

• For Reference: Basics of Matrices and Vectors
• Systems of ODEs as Models in Engineering Applications
• Basic Theory of Systems of ODEs. Wronskian
• Constant-Coefficient Systems. Phase Plane Method
• Criteria for Critical Points. Stability
• Qualitative Methods for Nonlinear Systems
• Nonhomogeneous Linear Systems of ODEs
• Sustainability and Ethical Considerations
• Chapter 4 Review Questions and Problems
• Summary of Chapter 4

CHAPTER 5 Series Solutions of ODEs. Special Functions

• Power Series Method
• egendre’s Equation. Legendre Polynomials P_n(x)
• Extended Power Series Method: Frobenius Method
• Bessel’s Equation. Bessel Functions J_v(x)
• Bessel Functions of the Y_v  (x). General Solution
• Sustainability and Ethical Considerations
• Chapter 5 Review Questions and Problems
• Summary of Chapter 5

CHAPTER 6 Laplace Transforms

• Laplace Transform. Linearity. First Shifting Theorem (s-Shifting)
• Transforms of Derivatives and Integrals. ODEs
• Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting)
• Short Impulses. Dirac’s Delta Function. Partial Fractions
• Convolution. Integral Equations
• Differentiation and Integration of Transforms. ODEs with Variable Coefficients
• Systems of ODEs
• Laplace Transform: General Formulas
• Table of Laplace Transforms
• Sustainability and Ethical Considerations
• Chapter 6 Review Questions and Problems
• Summary of Chapter 6

PART B Linear Algebra. Vector Calculus

CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems

• Matrices, Vectors: Addition and Scalar Multiplication
• Matrix Multiplication
• Linear Systems of Equations. Gauss Elimination
• Linear Independence. Rank of a Matrix. Vector Space
• Solutions of Linear Systems: Existence, Uniqueness
• For Reference: Second- and Third-Order Determinants
• Determinants. Cramer’s Rule
• Inverse of a Matrix. Gauss–Jordan Elimination
• Vector Spaces, Inner Product Spaces. Linear Transformations. Optional
• Sustainability and Ethical Considerations
• Chapter 7 Review Questions and Problems
• Summary of Chapter 7

CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems

• The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors
• Some Applications of Eigenvalue Problems
• Symmetric, Skew-Symmetric, and Orthogonal Matrices
• Eigenbases. Diagonalization. Quadratic Forms
• Complex Matrices and Forms. Optional
• Sustainability and Ethical Considerations
• Chapter 8 Review Questions and Problems
• Summary of Chapter 8

CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl

• Vectors in 2-Space and 3-Space
• Inner Product (Dot Product)
• Vector Product (Cross Product)
• Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives
• Curves. Arc Length. Curvature. Torsion
• Calculus Review: Functions of Several Variables. Optional
• Gradient of a Scalar Field. Directional Derivative
• Divergence of a Vector Field
• Curl of a Vector Field
• Sustainability and Ethical Considerations
• Chapter 9 Review Questions and Problems
• Summary of Chapter 9

CHAPTER 10 Vector Integral Calculus. Integral Theorems

• Line Integrals
• Path Independence of Line Integrals
• Calculus Review: Double Integrals. Optional
• Green’s Theorem in the Plane
• Surfaces for Surface Integrals
• Surface Integrals
• Triple Integrals. Divergence Theorem of Gauss
• Further Applications of the Divergence Theorem
• Stokes’s Theorem
• Sustainability and Ethical Considerations
• Chapter 10 Review Questions and Problems
• Summary of Chapter 10

PART C Fourier Analysis. Partial Differential Equations (PDEs)

CHAPTER 11 Fourier Analysis

• Fourier Series
• Arbitrary Period. Even and Odd Functions. Half-Range Expansions
• Forced Oscillations
• Approximation by Trigonometric Polynomials
• Sturm–Liouville Problems. Orthogonal Functions
• Orthogonal Series. Generalized Fourier Series
• Fourier Integral
• Fourier Cosine and Sine Transforms
• Fourier Transform. Discrete and Fast Fourier Transforms
• Tables of Transforms
• Sustainability and Ethical Considerations
• Chapter 11 Review Questions and Problems
• Summary of Chapter 11

CHAPTER 12 Partial Differential Equations (PDEs)

• Basic Concepts of PDEs
• Modeling: Vibrating String, Wave Equation
• Solution by Separating Variables. Use of Fourier Series
• D’Alembert’s Solution of the Wave Equation. Characteristics
• Modeling: Heat Flow from a Body in Space. Heat Equation
• Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem
• Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms
• Modeling: Membrane, Two-Dimensional Wave Equation
• Rectangular Membrane. Double Fourier Series
• Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series
• Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential
• Solution of PDEs by Laplace Transforms
• Sustainability and Ethical Considerations
• Chapter 12 Review Questions and Problems
• Summary of Chapter 12

PART D Complex Analysis

CHAPTER 13 Complex Numbers and Functions. Complex Differentiation

• Complex Numbers and Their Geometric Representation
• Polar Form of Complex Numbers. Powers and Roots
• Derivative. Analytic Function
• Cauchy–Riemann Equations. Laplace’s Equation
• Exponential Function
• Trigonometric and Hyperbolic Functions. Euler’s Formula
• Logarithm. General Power. Principal Value
• Sustainability and Ethical Considerations
• Chapter 13 Review Questions and Problems
• Summary of Chapter 13

CHAPTER 14 Complex Integration

• Line Integral in the Complex Plane
• Cauchy’s Integral Theorem
• Cauchy’s Integral Formula
• Derivatives of Analytic Functions
• Sustainability and Ethical Considerations
• Chapter 14 Review Questions and Problems
• Summary of Chapter 14

CHAPTER 15 Power Series, Taylor Series

• Sequences, Series, Convergence Tests
• Power Series
• Functions Given by Power Series
• Taylor and Maclaurin Series
• Uniform Convergence. Optional
• Sustainability and Ethical Considerations
• Chapter 15 Review Questions and Problems
• Summary of Chapter 15

CHAPTER 16 Laurent Series. Residue Integration

• Laurent Series
• Singularities and Zeros. Infinity
• Residue Integration Method
• Residue Integration of Real Integrals
• Sustainability and Ethical Considerations
• Chapter 16 Review Questions and Problems
• Summary of Chapter 16

CHAPTER 17 Conformal Mapping

• Geometry of Analytic Functions: Conformal Mapping
• Linear Fractional Transformations (Möbius Transformations)
• Special Linear Fractional Transformations
• Conformal Mapping by Other Functions
• Riemann Surfaces. Optional
• Sustainability and Ethical Considerations
• Chapter 17 Review Questions and Problems
• Summary of Chapter 17

CHAPTER 18 Complex Analysis and Potential Theory

• Electrostatic Fields
• Use of Conformal Mapping. Modeling
• Heat Problems
• Fluid Flow
• Poisson’s Integral Formula for Potentials
• General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem
• Sustainability and Ethical Considerations
• Chapter 18 Review Questions and Problems
• Summary of Chapter 18

PART E Numeric Analysis Software

CHAPTER 19 Numerics in General

• Introduction
• Solution of Equations by Iteration
• Interpolation
• Spline Interpolation
• Numeric Integration and Differentiation
• Sustainability and Ethical Considerations
• Chapter 19 Review Questions and Problems
• Summary of Chapter 19

CHAPTER 20 Numeric Linear Algebra

• Linear Systems: Gauss Elimination
• Linear Systems: LU-Factorization, Matrix Inversion
• Linear Systems: Solution by Iteration
• Linear Systems: Ill-Conditioning, Norms
• Least Squares Method
• Matrix Eigenvalue Problems: Introduction
• Inclusion of Matrix Eigenvalues
• Power Method for Eigenvalues
• Tridiagonalization and QR-Factorization
• Sustainability and Ethical Considerations
• Chapter 20 Review Questions and Problems
• Summary of Chapter 20

CHAPTER 21 Numerics for ODEs and PDEs

• Methods for First-Order ODEs
• Multistep Methods
• Methods for Systems and Higher Order ODEs
• Methods for Elliptic PDEs
• Neumann and Mixed Problems. Irregular Boundary
• Methods for Parabolic PDEs
• Method for Hyperbolic PDEs
• Sustainability and Ethical Considerations
• Chapter 21 Review Questions and Problems
• Summary of Chapter 21

PART F Optimization, Graphs

CHAPTER 22 Unconstrained Optimization. Linear Programming

• Basic Concepts. Unconstrained Optimization: Method of Steepest Descent
• Linear Programming
• Simplex Method
• Simplex Method: Difficulties
• Sustainability and Ethical Considerations
• Chapter 22 Review Questions and Problems
• Summary of Chapter 22

CHAPTER 23 Graphs. Combinatorial Optimization

• Graphs and Digraphs
• Shortest Path Problems. Complexity
• Bellman’s Principle. Dijkstra’s Algorithm
• Shortest Spanning Trees: Greedy Algorithm
• Shortest Spanning Trees: Prim’s Algorithm
• Flows in Networks
• Maximum Flow: Ford–Fulkerson Algorithm
• Bipartite Graphs. Assignment Problems
• Sustainability and Ethical Considerations
• Chapter 23 Review Questions and Problems
• Summary of Chapter 23

PART G Probability, Statistics (available online)

CHAPTER 24 Data Analysis. Probability Theory

• Data Representation. Average. Spread
• Experiments, Outcomes, Events
• Probability
• Permutations and Combinations
• Random Variables. Probability Distributions
• Mean and Variance of a Distribution
• Binomial, Poisson, and Hypergeometric Distributions
• Normal Distribution
• Distributions of Several Random Variables
• Sustainability and Ethical Considerations
• Chapter 24 Review Questions and Problems
• Summary of Chapter 24

CHAPTER 25 Mathematical Statistics

• Introduction. Random Sampling
• Point Estimation of Parameters
• Confidence Intervals
• Testing Hypotheses. Decisions
• Quality Control
• Acceptance Sampling
• Goodness of Fit. χ2-Test
• Nonparametric Tests
• Regression. Fitting Straight Lines. Correlation
• Sustainability and Ethical Considerations
• Chapter 25 Review Questions and Problems
• Summary of Chapter 25

PROJECTs

APPENDIX 1 References

APPENDIX 2 Answers to Odd-Numbered Problems (available online)

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

APPENDIX 6 Emerging Topics in Applied Mathematics