Advanced Engineering Mathematics – Erwin Kreyszig – 3rd Edition

Chapter 1: Ordinary Differential Equations of the First Order

1.1 Basic Concepts and Ideas
1.2 Geometrical Considerations. Isoclines
1.3 Separable Equations
1.4 Equations Reducible to Separable Form
1.5 Exact Differential Equations
1.6 Integrating Factors
1.7 Linear First-Order Differential Equations
1.8 Variation of Parameters
1.9 Electric Circuits
1.10 Families of Curves. Orthogonal Trajectories
1.11 Picard's Iteration Method
1.12 Existence and Uniqueness of Solutions

Chapter 2: Ordinary Linear Differential Equations

2.1 Homogeneous Linear Equations of the Second Order
2.2 Homogeneous Second-Order Equations with Constant Coefficients
2.3 General Solution. Fundamental System
2.4 Complex Roots of the Characteristic Equation. Initial Value Problem
2.5 Double Root of the Characteristic Equation
2.6 Free Oscillations
2.7 Cauchy Equation
2.8 Existence and Uniqueness of Solutions
2.9 Homogeneous Linear Equations of Arbitrary Order
2.10 Homogeneous Linear Equations of Arbitrary Order with Constant Coefficients
2.11 Nonhomogeneous Linear Equations
2.12 A Method for Solving Nonhomogeneous Linear Equations
2.13 Forced Oscillations. Resonance
2.14 Electric Circuits
2.15 Complex Method for Obtaining Particular Solutions
2.16 General Method for Solving Nonhomogeneous Equations

Chapter 3: Power Series Solutions of Differential Equations

3.1 The Power Series Method
3.2 Theoretical Basis of the Power Series Method
3.3 Legendre's Equation. Legendre Polynomials
3.4 Extended Power Series Method. Indicial Equation
3.5 Bessel's Equation. Bessel Functions of the First Kind
3.6 Bessel Functions of the Second Kind
3.7 Orthogonal Sets of Functions
3.8 Sturm-Liouville Problem
3.9 Orthogonality of Legendre Polynomials and Bessel Functions

Chapter 4: Laplace Transformation

4.1 Laplace Transform. Inverse Transform. Linearity
4.2 Laplace Transforms of Derivatives and Integrals
4.3 Transformation of Ordinary Differential Equations
4.4 Partial Fractions
4.5 Examples and Applications
4.6 Differentiation and Integration of Transforms
4.7 Unit Step Function
4.8 Shifting on the r-axis
4.9 Periodic Functions
4.10 Table of Some Laplace Transforms

Chapter 5: Linear Algebra Part I: Vectors

5.1 Scalars and Vectors
5.2 Components of a Vector
5.3 Addition of VectorsMultiplication by Scalars
5.4 Vector Spaces. Linear Dependence and Independence
5.5 Inner Product (Dot Product)
5.6 Inner Product Spaces
5.7 Vector Product (Cross Product)
5.8 Vector Products in Terms of Components
5.9 Scalar Triple Product. Other Repeated Products

Chapter 6: Linear Algebra Part II: Matrices and Determinants

6.1 Basic Concepts
6.2 Addition of MatricesMultiplication by Numbers
6.3 Transpose of a Matrix. Special Matrices
6.4 Matrix Multiplication
6.5 Systems of Linear Equations. Gauss Elimination
6.6 Determinants of Second and Third Order
6.7 Determinants of Arbitrary Order
6.8 Rank of a Matrix. Row Equivalence
6.9 Systems of Linear Equations: Existence and General Properties of Solutions
6.10 Linear Dependence and Rank. Singular Matrices
6.11 Systems of Linear Equations: Solution by Determinants
6.12 The Inverse of a Matrix
6.13 BilinearQuadraticHermitianand Skew-Hermitian Forms
6.14 EigenvaluesEigenvectors
6.15 Eigenvalues of HermitianSkew-Hermitianand Unitary Matrices

Chapter 7: Vector Differential Calculus. Vector Fields

7.1 Scalar Fields and Vector Fields
7.2 Vector Calculus
7.3 Curves
7.4 Arc Length
7.5 Tangent. Curvature and Torsion
7.6 Velocity and Acceleration
7.7 Chain Rule and Mean Value Theorem for Functions of Several Variables
7.8 Directional Derivative. Gradient of a Scalar Field
7.9 Transformation of Coordinate Systems and Vector Components
7.10 Divergence of a Vector Field
7.11 Curl of a Vector Field

Chapter 8: Line and Surface Integrals. Integral Theorems

8.1 Line Integral
8.2 Evaluation of Line Integrals
8.3 Double Integrals
8.4 Transformation of Double Integrals into Line Integrals,
8.5 Surfaces
8.6 Tangent Plane. First Fundamental Form. Area
8.7 Surface Integrals
8.8 Triple Integrals. Divergence Theorem of Gauss
8.9 Consequences and Applications of the Divergence Theorem
8.10 Stokes's Theorem
8.11 Consequences and Applications of Stolces's Theorem
8.12 Line Integrals Independent of Path

Chapter 9: Fourier Series and Integrals

9.1 Periodic Functions. Trigonometric Series
9.2 Fourier Series. Euler Formulas
9.3 Functions Having Arbitrary Period
9.4 Even and Odd Functions
9.5 Half-Range Expansions
9.6 Determination of Fourier Coefficients Without Integration
9.7 Forced Oscillations
9.8 Approximation by Trigonometric Polynomials. Square Error
9.9 The Fourier Integral

Chapter 10: Partial Differential Equations

10.1 Basic Concepts
10.2 Vibrating String. One-Dimensional Wave Equation
10.3 Separation of Variables (Product Method)
10.4 D'AIembert's Solution of the Wave Equation
10.5 One-Dimensional Heat Flow
10.6 Heat Flow in an Infinite Bar
10.7 Vibrating Membrane. Two-Dimensional Wave Equation
10.8 Rectangular Membrane
10.9 Laplacian in Polar Coordinates
10.10 Circular Membrane. Bessel's Equation
10.11 Laplace's Equation. Potential
10.12 Laplace's Equation in Spherical Coordinates. Legendre's Equation

Chapter 11: Complex Analytic Functions

11.1 Complex Numbers. Triangle Inequality
11.2 Limit. Derivative. Analytic Function
11.3 Cauchy-Riemann Equations. Laplace's Equation,
11.4 Rational Functions. Root
11.5 Exponential Function
11.6 Trigonometric and Hyperbolic Functions
11.7 Logarithm. General Power

Chapter 12: Conformal Mapping

12.1 Mapping
12.2 Conformal Mapping
12.3 Linear Fractional Transformations
12.4 Special Linear Fractional Transformations
12.5 Mapping by Other Elementary Functions
12.6 Riemann Surfaces

Chapter 13: Complex Integrals

13.1 Line Integral in the Complex Plane
13.2 Basic Properties of the Complex Line Integral
13.3 Cauchy's Integral Theorem
13.4 Evaluation of Line Integrals by Indefinite Integration
13.5 Cauchy's Integral Formula
13.6 The Derivatives of an Analytic Function

Chapter 14: Sequences and Series

14.1 Sequences
14.2 Series
14.3 Tests for Convergence and Divergence of Series
14.4 Operations on Series
14.5 Power Series
14.6 Functions Represented by Power Series

Chapter 15: Taylor and Laurent Series

15.1 Taylor Series
15.2 Taylor Series of Elementary Functions
15.3 Practical Methods for Obtaining Power Series
15.4 Uniform Convergence
15.5 Laurent Series
15.6 Behavior of Functions at Infinity,

Chapter 16: Integration by the Method of Residues

16.1 Zeros and Singularities
16.2 Residues
16.3 The Residue Theorem
16.4 Evaluation of Real Integrals

Chapter 17: Complex Analytic Functions and Potential Theory

17.1 Electrostatic Fields
17.2 Two-Dimensional Fluid Flow
17.3 General Properties of Harmonic Functions
17.4 Poisson's Integral Formula

Chapter 18: Numerical Analysis

18.1 Errors and Mistakes. Automatic Computers
18.2 Solution of Equations by Iteration
18.3 Finite Differences
18.4 Interpolation
18.5 Numerical Integration and Differentiation. 653
18.6 Numerical Methods for First-Order Differential Equations
18.7 Numerical Methods for Second-Order Differential Equations
18.8 Systems of Linear Equations. Gauss Elimination
18.9 Systems of Linear Equations. Solution by Iteration
18.10 Systems of Linear Equations. Ill-conditioning
18.11 Method of Least Squares
18.12 Inclusion of Matrix Eigenvalues
18.13 Determination of Eigenvalues by Iteration
18.14 Asymptotic Expansions

Chapter 19: Probability and Statistics

19.1 Nature and Purpose of Mathematical Statistics
19.2 Tabular and Graphical Representation of Samples
19.3 Sample Mean and Sample Variance
19.4 Random ExperimentsOutcomesEvents
19.5 Probability
19.6 Permutations and Combinations
19.7 Random Variables. Discrete and Continuous Distributions
19.8 Mean and Variance of a Distribution.
19.9 Binomial. Poissonand Hypergeometric Distributions
19.10 Normal Distribution
19.11 Distributions of Several Random Variables
19.12 Random Sampling. Random Numbers
19.13 Estimation of Parameters
19.14 Confidence Intervals
19.15 Testing of HypothesesDecisions
19.16 Quality Control
19.17 Acceptance Sampling
19.18 Goodness of Fit. ;(;2-Test
19.19 Nonparametric Tests
19.20 Pairs of Measurements. Fitting Straight Lines

Appendix 1 References
Appendix 2 Answers to Odd-Numbered Problems
Appendix 3 Some Formulas for Special Functions
Appendix 4 Tables
Index