Advanced Engineering Mathematics
10th Edition

Advanced Engineering Mathematics
Erwin Kreyszig

© 2011

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Topics covered

PART A Ordinary Differential Equations (ODEs)
CHAPTER 1 First-Order ODEs

  1. Basic Concepts. Modeling
  2. Geometric Meaning of y ƒ(x, y). Direction Fields, Euler’s Method
  3. Separable ODEs. Modeling
  4. Exact ODEs. Integrating Factors
  5. Linear ODEs. Bernoulli Equation. Population Dynamics
  6. Orthogonal Trajectories. Optional
  7. Existence and Uniqueness of Solutions for Initial Value Problems

CHAPTER 2 Second-Order Linear ODEs

  1. Homogeneous Linear ODEs of Second Order
  2. Homogeneous Linear ODEs with Constant Coefficients
  3. Differential Operators. Optional
  4. Modeling of Free Oscillations of a Mass–Spring System
  5. Euler–Cauchy Equations
  6. Existence and Uniqueness of Solutions. Wronskian
  7. Nonhomogeneous ODEs
  8. Modeling: Forced Oscillations. Resonance
  9. Modeling: Electric Circuits
  10. Solution by Variation of Parameters

CHAPTER 3 Higher Order Linear ODEs

  1. Homogeneous Linear ODEs
  2. Homogeneous Linear ODEs with Constant Coefficients
  3. Nonhomogeneous Linear ODEs

CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods

  1. For Reference: Basics of Matrices and Vectors
  2. Systems of ODEs as Models in Engineering Applications
  3. Basic Theory of Systems of ODEs. Wronskian
  4. Constant-Coefficient Systems. Phase Plane Method
  5. Criteria for Critical Points. Stability
  6. Qualitative Methods for Nonlinear Systems
  7. Nonhomogeneous Linear Systems of ODEs

CHAPTER 5 Series Solutions of ODEs. Special Functions

  1. Power Series Method
  2. Legendre's Equation. Legendre Polynomials Pn(x)
  3. Extended Power Series Method: Frobenius Method
  4. Bessel’s Equation. Bessel Functions (x)
  5. Bessel Functions of the Y (x). General Solution

CHAPTER 6 Laplace Transforms

  1. Laplace Transform. Linearity. First Shifting Theorem (s-Shifting)
  2. Transforms of Derivatives and Integrals. ODEs
  3. Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting)
  4. Short Impulses. Dirac's Delta Function. Partial Fractions
  5. Convolution. Integral Equations
  6. Differentiation and Integration of Transforms. ODEs with Variable Coefficients
  7. Systems of ODEs
  8. Laplace Transform: General Formulas
  9. Table of Laplace Transforms

PART B Linear Algebra. Vector Calculus

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

  1. Matrices, Vectors: Addition and Scalar Multiplication
  2. Matrix Multiplication
  3. Linear Systems of Equations. Gauss Elimination
  4. Linear Independence. Rank of a Matrix. Vector Space
  5. Solutions of Linear Systems: Existence, Uniqueness
  6. For Reference: Second- and Third-Order Determinants
  7. Determinants. Cramer’s Rule
  8. Inverse of a Matrix. Gauss–Jordan Elimination
  9. Vector Spaces, Inner Product Spaces. Linear Transformations. Optional

CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems

  1. The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors
  2. Some Applications of Eigenvalue Problems
  3. Symmetric, Skew-Symmetric, and Orthogonal Matrices
  4. Eigenbases. Diagonalisation. Quadratic Forms
  5. Complex Matrices and Forms. Optional 

CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl

  1. Vectors in 2-Space and 3-Space
  2. Inner Product (Dot Product)
  3. Vector Product (Cross Product)
  4. Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives
  5. Curves. Arc Length. Curvature. Torsion
  6. Calculus Review: Functions of Several Variables. Optional
  7. Gradient of a Scalar Field. Directional Derivative
  8. Divergence of a Vector Field
  9. Curl of a Vector Field

CHAPTER 10 Vector Integral Calculus. Integral Theorems

  1. Line Integrals
  2. Path Independence of Line Integrals
  3. Calculus Review: Double Integrals. Optional 
  4. Green’s Theorem in the Plane
  5. Surfaces for Surface Integrals
  6. Surface Integrals
  7. Triple Integrals. Divergence Theorem of Gauss
  8. Further Applications of the Divergence Theorem
  9. Stokes’s Theorem

PART C Fourier Analysis. Partial Differential Equations (PDEs)

CHAPTER 11 Fourier Analysis

  1. Fourier Series
  2. Arbitrary Period. Even and Odd Functions. Half-Range Expansions
  3. Forced Oscillations
  4. Approximation by Trigonometric Polynomials
  5. Sturm–Liouville Problems. Orthogonal Functions
  6. Orthogonal Series. Generalised Fourier Series
  7. Fourier Integral
  8. Fourier Cosine and Sine Transforms
  9. Fourier Transform. Discrete and Fast Fourier Transforms
  10. Tables of Transforms

CHAPTER 12 Partial Differential Equations (PDEs)

  1. Basic Concepts of PDEs
  2. Modeling: Vibrating String, Wave Equation
  3. Solution by Separating Variables. Use of Fourier Series
  4. D’Alembert’s Solution of the Wave Equation. Characteristics
  5. Modeling: Heat Flow from a Body in Space. Heat Equation
  6. Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem
  7. Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms
  8. Modeling: Membrane, Two-Dimensional Wave Equation
  9. Rectangular Membrane. Double Fourier Series
  10. Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series
  11. Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential
  12. Solution of PDEs by Laplace Transforms

PART D Complex Analysis

CHAPTER 13 Complex Numbers and Functions. Complex Differentiation

  1. Complex Numbers and Their Geometric Representation
  2. Polar Form of Complex Numbers. Powers and Roots
  3. Derivative. Analytic Function
  4. Cauchy–Riemann Equations. Laplace’s Equation
  5. Exponential Function
  6. Trigonometric and Hyperbolic Functions. Euler's Formula
  7. Logarithm. General Power. Principal Value

CHAPTER 14 Complex Integration

  1. Line Integral in the Complex Plane
  2. Cauchy's Integral Theorem
  3. Cauchy's Integral Formula
  4. Derivatives of Analytic Functions

CHAPTER 15 Power Series, Taylor Series

  1. Sequences, Series, Convergence Tests
  2. Power Series
  3. Functions Given by Power Series
  4. Taylor and Maclaurin Series
  5. Uniform Convergence. Optional

CHAPTER 16 Laurent Series. Residue Integration

  1. Laurent Series
  2. Singularities and Zeros. Infinity
  3. Residue Integration Method
  4. Residue Integration of Real Integrals 

CHAPTER 17 Conformal Mapping

  1. Geometry of Analytic Functions: Conformal Mapping
  2. Linear Fractional Transformations (Möbius Transformations)
  3. Special Linear Fractional Transformations
  4. Conformal Mapping by Other Functions
  5. Riemann Surfaces. Optional

CHAPTER 18 Complex Analysis and Potential Theory

  1. Electrostatic Fields
  2. Use of Conformal Mapping. Modeling
  3. Heat Problems
  4. Fluid Flow
  5. Poisson's Integral Formula for Potentials
  6. General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem

PART E Numeric Analysis


CHAPTER 19 Numerics in General

  1. Introduction
  2. Solution of Equations by Iteration
  3. Interpolation
  4. Spline Interpolation
  5. Numeric Integration and Differentiation

CHAPTER 20 Numeric Linear Algebra

  1. Linear Systems: Gauss Elimination
  2. Linear Systems: LU-Factorisation, Matrix Inversion
  3. Linear Systems: Solution by Iteration
  4. Linear Systems: Ill-Conditioning, Norms
  5. Least Squares Method
  6. Matrix Eigenvalue Problems: Introduction
  7. Inclusion of Matrix Eigenvalues
  8. Power Method for Eigenvalues
  9. Tridiagonalisation and QR-Factorisation

CHAPTER 21 Numerics for ODEs and PDEs

  1. Methods for First-Order ODEs
  2. Multistep Methods
  3. Methods for Systems and Higher Order ODEs
  4. Methods for Elliptic PDEs
  5. Neumann and Mixed Problems. Irregular Boundary
  6. Methods for Parabolic PDEs
  7. Method for Hyperbolic PDEs

PART F Optimisation, Graphs

CHAPTER 22 Unconstrained Optimisation. Linear Programming

  1. Basic Concepts. Unconstrained Optimisation: Method of Steepest Descent
  2. Linear Programming
  3. Simplex Method
  4. Simplex Method: Difficulties

CHAPTER 23 Graphs. Combinatorial Optimisation

  1. Graphs and Digraphs
  2. Shortest Path Problems. Complexity
  3. Bellman's Principle. Dijkstra’s Algorithm
  4. Shortest Spanning Trees: Greedy Algorithm 
  5. Shortest Spanning Trees: Prim’s Algorithm
  6. Flows in Networks
  7. Maximum Flow: Ford–Fulkerson Algorithm
  8. Bipartite Graphs. Assignment Problems

APPENDIX 1 References A1

APPENDIX 2 Answers to Selected Problems A4

APPENDIX 3 Auxiliary Material A51

A3.1 Formulas for Special Functions A51

A3.2 Partial Derivatives A57

A3.3 Sequences and Series A60

A3.4 Grad, Div, Curl, 2 in Curvilinear Coordinates A62

APPENDIX 4 Additional Proofs A65

APPENDIX 5 Tables A85