ISBN13: 978-0534552084

ISBN10: 0534552080

Cover type:

Edition/Copyright: 6TH 07

Publisher: Thomson Learning

Published: 2007

International: No

ISBN10: 0534552080

Cover type:

Edition/Copyright: 6TH 07

Publisher: Thomson Learning

Published: 2007

International: No

Through previous editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. Advanced Engineering Mathematics features a greater number of examples and problems and is fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts and problem sets, incorporating the use of leading software packages. Computational assistance, exercises and projects have been included to encourage students to make use of these computational tools. The content is organized into eight parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations and Qualitative Methods, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, Partial Differential Equations, Complex Analysis, and Probability and Statistics.

**Benefits:**

- NEW! Chapter 5: Numerical Approximation of Solutions (including material from Chapter 5 of the 4e).
- NEW! Chapter 26: Counting and Probability.
- NEW! Chapter 27: Statistics.
- A new Part VIII: Probability and Statistics.
- New material on numerical approximation of the wave and heat equations in Chapters 17 & 18.
- Reduced page count.

**Part I - Ordinary Differential Equations **

Chapter 1 - First Order Differential Equations

1.1 Preliminary Concepts

1.1.1 General and Particular Solutions

1.1.2 Implicitly Defined Solutions

1.1.3 Integral Curves

1.1.4 The Initial Value Problem

1.1.5 Direction Fields

1.2 Separable Equations

1.2.1 Some Applications of Separable Differential Equations

1.3 Linear Differential Equations

1.4 Exact Differential Equations

1.5 Integrating Factors

1.5.1 Separable Equations and Integrating Factors

1.5.2 Linear Equations and Integrating Factors

1.6 Homogeneous, Bernoulli and Riccati Equations

1.6.1 Homogeneous Differential Equations

1.6.2 The Bernoulli Equation

1.6.3 The Riccati Equation

1.7 Applications to Mechanics, Electrical Circuits and Orthogonal Trajectories

1.7.1 Mechanics

1.7.2 Electrical Circuits

1.7.3 Orthogonal Trajectories

1.8 Existence and Uniqueness for Solutions of Initial Value Problems

Chapter 2 - Linear Second Order Differential Equations

2.1 Preliminary Concepts

2.2 Theory of Solutions of y′′ + p(x)y′ + q(x)y = f(x)

2.2.1 The Homogeneous Equation y′′ + p(x)y′ + q(x) = 0

2.2.2 The Nonhomogeneous Equation y′′ + p(x)y′ + q(x)y = f(x)

2.3 Reduction of Order

2.4 The Constant Coefficient Homogeneous Linear Equation

2.4.1 Case 1 A2 - 4B > 0

2.4.2 Case 2 A2 - 4B = 0

2.4.3 Case 3 A2 - 4B < 0

2.4.4 An Alternative General Solution In the Complex Root Case

2.5 Euler's Equation

2.6 The Nonhomogeneous Equation y′′ + p(x)y′ + q(x)y = f(x)

2.6.1 The Method of Variation of Parameters

2.6.2 The Method of Undetermined Coefficients

2.6.3 The Principle of Superposition

2.6.4 Higher Order Differential equations

2.7 Application of Second Order Differential Equations to a Mechanical System

2.7.1 Unforced Motion

2.7.2 Forced Motion

2.7.3 Resonance

2.7.4 Beats

2.7.5 Analogy With An Electrical Circuit

Chapter 3 - The Laplace Transform

3.1 Definition and Basic Properties

3.2 Solution of Initial Value Problems Using the Laplace Transform

3.3 Shifting Theorems and the Heaviside Function

3.3.1 The First Shifting Theorem

3.3.2 The Heaviside Function and Pulses

3.3.3 The Second Shifting Theorem

3.3.4 Analysis of Electrical Circuits

3.4 Convolution

3.5 Unit Impulses and the Dirac Delta Function

3.6 Laplace Transform Solution of Systems

3.7 Differential Equations With Polynomial Coefficients

Chapter 4 - Series Solutions

4.1 Power Series Solutions of Initial Value Problems

4.2 Power Series Solutions Using Recurrence Relations

4.3 Singular Points and the Method of Frobenius

4.4 Second Solutions and Logarithm Factors

4.5 Appendix on Power Series

4.5.1 Convergence of Power Series

4.5.2 Algebra and Calculus of Power Series

4.5.3 Taylor and Maclaurin Expansions

4.5.4 Shifting Indices

Chapter 5 - Numerical Approximation of Solutions

5.1 Euler's Method

5.1.1 A Problem in Radioactive Waste Disposal

5.2 One-Step Methods

5.2.1 The Second Order Taylor Method

5.2.2 The Modified Euler Method

5.2.3 Runge-Kutta Methods

5.3 Multistep Methods

5.3.1 Multistep Methods

**Part II - Vectors and Linear Algebra **

Chapter 6 - Vectors and Vector Spaces

6.1 The Algebra and Geometry of Vectors

6.2 The Dot Product

6.3 The Cross Product

6.4 The Vector Space Rⁿ

6.5 Linear Independence, Spanning Sets and Dimension in Rⁿ

6.6 Abstract Vector Spaces

Chapter 7 - Matrices and Systems of Linear Equations

7.1 Matrices

7.1.1 Matrix Algebra

7.1.2 Matrix Notation for Systems of Linear Equations

7.1.3 Some Special Matrices

7.1.4 Another Rationale for the Definition of Matrix Multiplication

7.1.5 Random Walks in Crystals

7.2 Elementary Row Operations and Elementary Matrices

7.3 The Row Echelon Form of a Matrix

7.4 The Row and Column Spaces of a Matrix and Rank of a Matrix

7.5 Solution of Homogeneous Systems of Linear Equations

7.6 The Solution Space of AX = O

7.7 Nonhomogeneous Systems of Linear Equations

7.7.1 The Structure of Solutions of AX = B

7.7.2 Existence and Uniqueness of Solutions of AX = B

7.8 Summary for Linear Systems

7.9 Matrix Inverses

7.9.1 A Method for Finding A-1

Chapter 8 - Determinants

8.1 Permutations

8.2 Definition of the Determinant

8.3 Properties of Determinants

8.4 Evaluation of Determinants by Elementary Row and Column Operations

8.5 Cofactor Expansions

8.6 Determinants of Triangular Matrices

8.7 A Determinant Formula for a Matrix Inverse

8.8 Cramer's Rule

8.9 The Matrix Tree Theorem

Chapter 9 - Eigenvalues, Diagonalization and Special Matrices

9.1 Eigenvalues and Eigenvectors

9.1.1 Gerschgorin's Theorem

9.2 Diagonalization of Matrices

9.3 Orthogonal and Symmetric Matrices

9.4 Quadratic Forms

9.5 Unitary, Hermitian and Skew Hermitian Matrices

**Part III - Systems of Differential Equations and Qualitative Methods **

Chapter 10 - Systems of Linear Differential Equations

10.1 Theory of Systems of Linear First Order Differential Equations

10.1.1 Theory of the Homogeneous System X′ = AX

10.1.2 General Solution of the Nonhomogeneous System X′ = AX + G

10.2 Solution of X′ = AX When A Is Constant

10.2.1 Solution of X′ = AX When A Has Complex Eigenvalues

10.2.2 Solution of X′ = AX When A Does Not Have n Linearly Independent Eigenvectors

10.2.3 Solution of X′ = AX By Diagonalizing A

10.2.4 Exponential Matrix Solutions of X0 = AX

10.3 Solution of X′ = AX + G

10.3.1 Variation of Parameters

10.3.2 Solution of X′ = AX + G By Diagonalizing A

Chapter 11 - Qualitative Methods and Systems of Nonlinear Differential Equations

11.1 Nonlinear Systems and Existence of Solutions

11.2 The Phase Plane, Phase Portraits and Direction Fields

11.3 Phase Portraits of Linear Systems

11.4 Critical Points and Stability

11.5 Almost Linear Systems

11.6 Predator/Prey Population Models

11.6.1 A Simple Predator/Prey Model

11.6.2 An Extended Predator/Prey Model

11.7 Competing Species Models

11.7.1 A Simple Competing Species Model

11.7.2 An Extended Competing Species Model

11.8 Lyapunov's Stability Criteria

11.9 Limit Cycles and Periodic Solutions

**Part IV - Vector Analysis **

Chapter 12 - Vector Differential Calculus

12.1 Vector Functions of One Variable

12.2 Velocity, Acceleration, Curvature and Torsion

12.2.1 Tangential and Normal Components of Acceleration

12.2.2 Curvature As a Function of t

12.2.3 The Frenet Formulas

12.3 Vector Fields and Streamlines

12.4 The Gradient Field and Directional Derivatives

12.4.1 Level Surfaces, Tangent Planes and Normal Lines

12.5 Divergence and Curl

12.5.1 A Physical Interpretation of Divergence

12.5.2 A Physical Interpretation of Curl

Chapter 13 - Vector Integral Calculus

13.1 Line Integrals

13.1.1 Line Integral With Respect to Arc Length

13.2 Green's Theorem

13.2.1 An Extension of Green's Theorem

13.3 Independence of Path and Potential Theory In the Plane

13.3.1 A More Critical Look at Theorem 13.5

13.4 Surfaces In 3- Space and Surface Integrals

13.4.1 Normal Vector to a Surface

13.4.2 The Tangent Plane to a Surface

13.4.3 Smooth and Piecewise Smooth Surfaces

13.4.4 Surface Integrals

13.5 Applications of Surface Integrals

13.5.1 Surface Area

13.5.2 Mass and Center of Mass of a Shell

13.5.3 Flux of a Vector Field Across a Surface

13.6 Preparation for the Integral Theorems of Gauss and Stokes

13.7 The Divergence Theorem of Gauss

13.7.1 Archimedes's Principle

13.7.2 The Heat Equation

13.7.3 The Divergence Theorem As A Conservation of Mass Principle

13.7.4 Green's Identities

13.8 The Integral Theorem of Stokes

13.8.1 An Interpretation of Curl

13.8.2 Potential Theory in 3- Space

**Part V - Fourier Analysis, Orthogonal Expansions and Wavelets **

Chapter 14 - Fourier Series

14.1 Why Fourier Series?

14.2 The Fourier Series of a Function

14.2.1 Even and Odd Functions

14.3 Convergence of Fourier Series

14.3.1 Convergence at the End Points

14.3.2 A Second Convergence Theorem

14.3.3 Partial Sums of Fourier Series

14.3.4 The Gibbs Phenomenon

14.4 Fourier Cosine and Sine Series

14.4.1 The Fourier Cosine Series of a Function

14.4.2 The Fourier Sine Series of a Function

14.5 Integration and Differentiation of Fourier Series

14.6 The Phase Angle Form of a Fourier Series

14.7 Complex Fourier Series and the Frequency Spectrum

14.7.1 Review of Complex Numbers

14.7.2 Complex Fourier Series

Chapter 15 - The Fourier Integral and Fourier Transforms

15.1 The Fourier Integral

15.2 Fourier Cosine and Sine Integrals

15.3 The Complex Fourier Integral and the Fourier Transform

15.4 Additional Properties and Applications of the Fourier Transform

15.4.1 The Fourier Transform of a Derivative

15.4.2 Frequency Differentiation

15.4.3 The Fourier Transform of an Integral

15.4.4 Convolution

15.4.5 Filtering and the Dirac Delta Function

15.4.6 The Windowed Fourier Transform

15.4.7 The Shannon Sampling Theorem

15.4.8 Lowpass and Bandpass Filters

15.5 The Fourier Cosine and Sine Transforms

15.6 The Finite Fourier Cosine and Sine Transforms

15.7 The Discrete Fourier Transform

15.7.1 Linearity and Periodicity

15.7.2 The Inverse N- Point DFT

15.7.3 DFT Approximation of Fourier Coefficients

15.8 Sampled Fourier Series

15.8.1 Approximation of a Fourier Transform by an N- Point DFT

15.8.2 Filtering

15.9 The Fast Fourier Transform

15.9.1 Computational Efficiency of the FFT

15.9.2 Use of the FFT in Analyzing Power Spectral Densities of Signals

15.9.3 Filtering Noise From a Signal

15.9.4 Analysis of the Tides in Morro Bay

Chapter 16 - Special Functions, Orthogonal Expansions and Wavelets

16.1 Legendre Polynomials

16.1.1 A Generating Function for the Legendre Polynomials

16.1.2 A Recurrence Relation for the Legendre Polynomials

16.1.3 Orthogonality of the Legendre Polynomials

16.1.4 Fourier-Legendre Series

16.1.5 Computation of Fourier-Legendre Coefficients

16.1.6 Zeros of the Legendre Polynomials

16.1.7 Derivative and Integral Formulas for Pn(x)

16.2 Bessel Functions

16.2.1 The Gamma Function

16.2.2 Bessel Functions of the First Kind and Solutions of Bessel's Equation

16.2.3 Bessel Functions of the Second Kind

16.2.4 Modified Bessel Functions

16.2.5 Some Applications of Bessel Functions

16.2.6 A Generating Function for Jn(x)

16.2.7 An Integral Formula for Jn(x)

16.2.8 A Recurrence Relation for Jv(x)

16.2.9 Zeros of Jv(x)

16.2.10Fourier-Bessel Expansions

16.2.11Fourier-Bessel Coefficients

16.3 Sturm-Liouville Theory and Eigenfunction Expansions

16.3.1 The Sturm-Liouville Problem

16.3.2 The Sturm-Liouville Theorem

16.3.3 Eigenfunction Expansions

16.3.4 Approximation In the Mean and Bessel's Inequality

16.3.5 Convergence in the Mean and Parseval's Theorem

16.3.6 Completeness of the Eigenfunctions

16.4 Orthogonal Polynomials

16.4.1 Chebyshev Polynomials

16.4.2 Laguerre Polynomials

16.4.3 Hermite Polynomials

16.5 Wavelets

16.5.1 The Idea Behind Wavelets

16.5.2 The Haar Wavelets

16.5.3 A Wavelet Expansion

16.5.4 Multiresolution Analysis With Haar Wavelets

16.5.5 General Construction of Wavelets and Multiresolution Analysis

16.5.6 Shannon Wavelets

**Part VI - Partial Differential Equations **

Chapter 17 - The Wave Equation

17.1 The Wave Equation and Initial and Boundary Conditions

17.2 Fourier Series Solutions of the Wave Equation

17.2.1 Vibrating String With Zero Initial Velocity

17.2.2 Vibrating String with Given Initial Velocity and Zero Initial Displacement

17.2.3 Vibrating String With Initial Displacement and Velocity

17.2.4 Verification of Solutions

17.2.5 Transformation of Boundary Value Problems Involving the Wave Equation

17.2.6 Effects of Initial Conditions and Constants on the Motion

17.2.7 Numerical Solution of the Wave Equation

17.3 Wave Motion Along Infinite and Semi-Infinite Strings

17.3.1 Wave Motion Along an Infinite String

17.3.2 Wave Motion Along a Semi-Infinite String

17.3.3 Fourier Transform Solution of Problems on Unbounded Domains

17.4 Characteristics and d'Alembert's Solution

17.4.1 A Nonhomogeneous Wave Equation

17.4.2 Forward and Backward Waves

17.5 Normal Modes of Vibration of a Circular Elastic Membrane

17.6 Vibrations of a Circular Elastic Membrane, Revisited

17.7 Vibrations of a Rectangular Membrane

Chapter 18 - The Heat Equation

18.1 The Heat Equation and Initial and Boundary Conditions

18.2 Fourier Series Solutions of the Heat Equation

18.2.1 Ends of the Bar Kept at Temperature Zero

18.2.2 Temperature in a Bar With Insulated Ends

18.2.3 Temperature Distribution in a Bar With Radiating End

18.2.4 Transformations of Boundary Value Problems Involving the Heat Equation

18.2.5 A Nonhomogeneous Heat Equation

18.2.6 Effects of Boundary Conditions and Constants on Heat Conduction

18.2.7 Numerical Approximation of Solutions

18.3 Heat Conduction in Infinite Media

18.3.1 Heat Conduction in an Infinite Bar

18.3.2 Heat Conduction in a Semi-Infinite Bar

18.3.3 Integral Transform Methods for the Heat Equation in an Infinite Medium

18.4 Heat Conduction in an Infinite Cylinder

18.5 Heat Conduction in a Rectangular Plate

Chapter 19 - The Potential Equation

19.1 Harmonic Functions and the Dirichlet Problem

19.2 Dirichlet Problem for a Rectangle

19.3 Dirichlet Problem for a Disk

19.4 Poisson's Integral Formula for the Disk

19.5 Dirichlet Problems in Unbounded Regions

19.5.1 Dirichlet Problem for the Upper Half Plane

19.5.2 Dirichlet Problem for the Right Quarter Plane

19.5.3 An Electrostatic Potential Problem

19.6 A Dirichlet Problem for a Cube

19.7 The Steady-State Heat Equation for a Solid Sphere

19.8 The Neumann Problem

19.8.1 A Neumann Problem for a Rectangle

19.8.2 A Neumann Problem for a Disk

19.8.3 A Neumann Problem for the Upper Half Plane

**Part VII - Complex Analysis **

Chapter 20 - Geometry and Arithmetic of Complex Numbers

20.1 Complex Numbers

20.1.1 The Complex Plane

20.1.2 Magnitude and Conjugate

20.1.3 Complex Division

20.1.4 Inequalities

20.1.5 Argument and Polar Form of a Complex Number

20.1.6 Ordering

20.1.7 Binomial Expansion of (z + w)n

20.2 Loci and Sets of Points in the Complex Plane

20.2.1 Distance

20.2.2 Circles and Disks

20.2.3 The Equation |z - a| = |z - b|

20.2.4 Other Loci

20.2.5 Interior Points, Boundary Points, and Open and Closed Sets

20.2.6 Limit Points

20.2.7 Complex Sequences

20.2.8 Subsequences

20.2.9 Compactness and the Bolzano-Weierstrass Theorem

Chapter 21 - Complex Functions

21.1 Limits, Continuity and Derivatives

21.1.1 Limits

21.1.2 Continuity

21.1.3 The Derivative of a Complex Function

21.1.4 The Cauchy-Riemann Equations

21.2 Power Series

21.2.1 Series of Complex Numbers

21.2.2 Power Series

21.3 The Exponential and Trigonometric Functions

21.4 The Complex Logarithm

21.5 Powers

21.5.1 Integer Powers

21.5.2 z1/n for Positive Integer n

21.5.3 Rational Powers

21.5.4 Powers zw

Chapter 22 - Complex Integration

22.1 Curves in the Plane

22.2 The Integral of a Complex Function

22.2.1 The Complex Integral in Terms of Real Integrals

22.2.2 Properties of Complex Integrals

22.2.3 Integrals of Series of Functions

22.3 Cauchy's Theorem

22.3.1 Proof of Cauchy's Theorem for a Special Case

22.3.2 Proof of Cauchy's Theorem for a Rectangle

22.4 Consequences of Cauchy's Theorem

22.4.1 Independence of Path

22.4.2 The Deformation Theorem

22.4.3 Cauchy's Integral Formula

22.4.4 Cauchy's Integral Formula for Higher Derivatives

22.4.5 Bounds on Derivatives and Liouville's Theorem

22.4.6 An Extended Deformation Theorem

Chapter 23 - Series Representations of Functions

23.1 Power Series Representations

23.1.1 Isolated Zeros and the Identity Theorem

23.1.2 The Maximum Modulus Theorem

23.2 The Laurent Expansion

Chapter 24 - Singularities and the Residue Theorem

24.1 Singularities

24.2 The Residue Theorem

24.3 Some Applications of the Residue Theorem

24.3.1 The Argument Principle

24.3.2 Rouché's Theorem

24.3.3 Summation of Real Series

24.3.4 An Inversion Formula for the Laplace Transform

24.3.5 Evaluation of Real Integrals

Chapter 25 - Conformal Mappings

25.1 Functions as Mappings

25.2 Conformal Mappings

25.2.1 Linear Fractional Transformations

25.3 Construction of Conformal Mappings Between Domains

25.3.1 Schwarz-Christoffel Transformation

25.4 Harmonic Functions and the Dirichlet Problem

25.4.1 Solution of Dirichlet Problems by Conformal Mapping

25.5 Complex Function Models of Plane Fluid Flow

**Part VIII - Probability and Statistics **

Chapter 26 - Counting and Probability

26.1 The Multiplication Principle

26.2 Permutations

26.3 Choosing r Objects From n Objects

26.3.1 r Objects From n Objects, With Order

26.3.2 r Objects From n Objects, Without Order

26.3.3 Tree Diagrams

26.4 Events and Sample Spaces

26.5 The Probability of an Event

26.6 Complementary Events

26.7 Conditional Probability

26.8 Independent Events

26.8.1 The Product Rule

26.9 Tree Diagrams In Computing Probabilities

26.10 Bayes' Theorem

26.11 Expected Value

Chapter 27 - Statistics

27.1 Measures of Center and Variation

27.1.1 Measures of Center

27.1.2 Measures of Variation

27.2 Random Variables and Probability Distributions

27.3 The Binomial and Poisson Distributions

27.3.1 The Binomial Distribution

27.3.2 The Poisson Distribution

27.4 A Coin Tossing Experiment, Normally Distributed Data, and the Bell Curve

27.4.1 The Standard Bell Curve

27.4.2 The 68, 95, 99.7 Rule

27.5 Sampling Distributions and the Central Limit Theorem

27.6 Confidence Intervals and Estimating Population Proportion

27.7 Estimating Population Mean and the Student t Distribution

27.8 Correlation and Regression

ISBN10: 0534552080

Cover type:

Edition/Copyright: 6TH 07

Publisher: Thomson Learning

Published: 2007

International: No

Through previous editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. Advanced Engineering Mathematics features a greater number of examples and problems and is fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts and problem sets, incorporating the use of leading software packages. Computational assistance, exercises and projects have been included to encourage students to make use of these computational tools. The content is organized into eight parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations and Qualitative Methods, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, Partial Differential Equations, Complex Analysis, and Probability and Statistics.

**Benefits:**

- NEW! Chapter 5: Numerical Approximation of Solutions (including material from Chapter 5 of the 4e).
- NEW! Chapter 26: Counting and Probability.
- NEW! Chapter 27: Statistics.
- A new Part VIII: Probability and Statistics.
- New material on numerical approximation of the wave and heat equations in Chapters 17 & 18.
- Reduced page count.

Table of Contents

**Part I - Ordinary Differential Equations **

Chapter 1 - First Order Differential Equations

1.1 Preliminary Concepts

1.1.1 General and Particular Solutions

1.1.2 Implicitly Defined Solutions

1.1.3 Integral Curves

1.1.4 The Initial Value Problem

1.1.5 Direction Fields

1.2 Separable Equations

1.2.1 Some Applications of Separable Differential Equations

1.3 Linear Differential Equations

1.4 Exact Differential Equations

1.5 Integrating Factors

1.5.1 Separable Equations and Integrating Factors

1.5.2 Linear Equations and Integrating Factors

1.6 Homogeneous, Bernoulli and Riccati Equations

1.6.1 Homogeneous Differential Equations

1.6.2 The Bernoulli Equation

1.6.3 The Riccati Equation

1.7 Applications to Mechanics, Electrical Circuits and Orthogonal Trajectories

1.7.1 Mechanics

1.7.2 Electrical Circuits

1.7.3 Orthogonal Trajectories

1.8 Existence and Uniqueness for Solutions of Initial Value Problems

Chapter 2 - Linear Second Order Differential Equations

2.1 Preliminary Concepts

2.2 Theory of Solutions of y′′ + p(x)y′ + q(x)y = f(x)

2.2.1 The Homogeneous Equation y′′ + p(x)y′ + q(x) = 0

2.2.2 The Nonhomogeneous Equation y′′ + p(x)y′ + q(x)y = f(x)

2.3 Reduction of Order

2.4 The Constant Coefficient Homogeneous Linear Equation

2.4.1 Case 1 A2 - 4B > 0

2.4.2 Case 2 A2 - 4B = 0

2.4.3 Case 3 A2 - 4B < 0

2.4.4 An Alternative General Solution In the Complex Root Case

2.5 Euler's Equation

2.6 The Nonhomogeneous Equation y′′ + p(x)y′ + q(x)y = f(x)

2.6.1 The Method of Variation of Parameters

2.6.2 The Method of Undetermined Coefficients

2.6.3 The Principle of Superposition

2.6.4 Higher Order Differential equations

2.7 Application of Second Order Differential Equations to a Mechanical System

2.7.1 Unforced Motion

2.7.2 Forced Motion

2.7.3 Resonance

2.7.4 Beats

2.7.5 Analogy With An Electrical Circuit

Chapter 3 - The Laplace Transform

3.1 Definition and Basic Properties

3.2 Solution of Initial Value Problems Using the Laplace Transform

3.3 Shifting Theorems and the Heaviside Function

3.3.1 The First Shifting Theorem

3.3.2 The Heaviside Function and Pulses

3.3.3 The Second Shifting Theorem

3.3.4 Analysis of Electrical Circuits

3.4 Convolution

3.5 Unit Impulses and the Dirac Delta Function

3.6 Laplace Transform Solution of Systems

3.7 Differential Equations With Polynomial Coefficients

Chapter 4 - Series Solutions

4.1 Power Series Solutions of Initial Value Problems

4.2 Power Series Solutions Using Recurrence Relations

4.3 Singular Points and the Method of Frobenius

4.4 Second Solutions and Logarithm Factors

4.5 Appendix on Power Series

4.5.1 Convergence of Power Series

4.5.2 Algebra and Calculus of Power Series

4.5.3 Taylor and Maclaurin Expansions

4.5.4 Shifting Indices

Chapter 5 - Numerical Approximation of Solutions

5.1 Euler's Method

5.1.1 A Problem in Radioactive Waste Disposal

5.2 One-Step Methods

5.2.1 The Second Order Taylor Method

5.2.2 The Modified Euler Method

5.2.3 Runge-Kutta Methods

5.3 Multistep Methods

5.3.1 Multistep Methods

**Part II - Vectors and Linear Algebra **

Chapter 6 - Vectors and Vector Spaces

6.1 The Algebra and Geometry of Vectors

6.2 The Dot Product

6.3 The Cross Product

6.4 The Vector Space Rⁿ

6.5 Linear Independence, Spanning Sets and Dimension in Rⁿ

6.6 Abstract Vector Spaces

Chapter 7 - Matrices and Systems of Linear Equations

7.1 Matrices

7.1.1 Matrix Algebra

7.1.2 Matrix Notation for Systems of Linear Equations

7.1.3 Some Special Matrices

7.1.4 Another Rationale for the Definition of Matrix Multiplication

7.1.5 Random Walks in Crystals

7.2 Elementary Row Operations and Elementary Matrices

7.3 The Row Echelon Form of a Matrix

7.4 The Row and Column Spaces of a Matrix and Rank of a Matrix

7.5 Solution of Homogeneous Systems of Linear Equations

7.6 The Solution Space of AX = O

7.7 Nonhomogeneous Systems of Linear Equations

7.7.1 The Structure of Solutions of AX = B

7.7.2 Existence and Uniqueness of Solutions of AX = B

7.8 Summary for Linear Systems

7.9 Matrix Inverses

7.9.1 A Method for Finding A-1

Chapter 8 - Determinants

8.1 Permutations

8.2 Definition of the Determinant

8.3 Properties of Determinants

8.4 Evaluation of Determinants by Elementary Row and Column Operations

8.5 Cofactor Expansions

8.6 Determinants of Triangular Matrices

8.7 A Determinant Formula for a Matrix Inverse

8.8 Cramer's Rule

8.9 The Matrix Tree Theorem

Chapter 9 - Eigenvalues, Diagonalization and Special Matrices

9.1 Eigenvalues and Eigenvectors

9.1.1 Gerschgorin's Theorem

9.2 Diagonalization of Matrices

9.3 Orthogonal and Symmetric Matrices

9.4 Quadratic Forms

9.5 Unitary, Hermitian and Skew Hermitian Matrices

**Part III - Systems of Differential Equations and Qualitative Methods **

Chapter 10 - Systems of Linear Differential Equations

10.1 Theory of Systems of Linear First Order Differential Equations

10.1.1 Theory of the Homogeneous System X′ = AX

10.1.2 General Solution of the Nonhomogeneous System X′ = AX + G

10.2 Solution of X′ = AX When A Is Constant

10.2.1 Solution of X′ = AX When A Has Complex Eigenvalues

10.2.2 Solution of X′ = AX When A Does Not Have n Linearly Independent Eigenvectors

10.2.3 Solution of X′ = AX By Diagonalizing A

10.2.4 Exponential Matrix Solutions of X0 = AX

10.3 Solution of X′ = AX + G

10.3.1 Variation of Parameters

10.3.2 Solution of X′ = AX + G By Diagonalizing A

Chapter 11 - Qualitative Methods and Systems of Nonlinear Differential Equations

11.1 Nonlinear Systems and Existence of Solutions

11.2 The Phase Plane, Phase Portraits and Direction Fields

11.3 Phase Portraits of Linear Systems

11.4 Critical Points and Stability

11.5 Almost Linear Systems

11.6 Predator/Prey Population Models

11.6.1 A Simple Predator/Prey Model

11.6.2 An Extended Predator/Prey Model

11.7 Competing Species Models

11.7.1 A Simple Competing Species Model

11.7.2 An Extended Competing Species Model

11.8 Lyapunov's Stability Criteria

11.9 Limit Cycles and Periodic Solutions

**Part IV - Vector Analysis **

Chapter 12 - Vector Differential Calculus

12.1 Vector Functions of One Variable

12.2 Velocity, Acceleration, Curvature and Torsion

12.2.1 Tangential and Normal Components of Acceleration

12.2.2 Curvature As a Function of t

12.2.3 The Frenet Formulas

12.3 Vector Fields and Streamlines

12.4 The Gradient Field and Directional Derivatives

12.4.1 Level Surfaces, Tangent Planes and Normal Lines

12.5 Divergence and Curl

12.5.1 A Physical Interpretation of Divergence

12.5.2 A Physical Interpretation of Curl

Chapter 13 - Vector Integral Calculus

13.1 Line Integrals

13.1.1 Line Integral With Respect to Arc Length

13.2 Green's Theorem

13.2.1 An Extension of Green's Theorem

13.3 Independence of Path and Potential Theory In the Plane

13.3.1 A More Critical Look at Theorem 13.5

13.4 Surfaces In 3- Space and Surface Integrals

13.4.1 Normal Vector to a Surface

13.4.2 The Tangent Plane to a Surface

13.4.3 Smooth and Piecewise Smooth Surfaces

13.4.4 Surface Integrals

13.5 Applications of Surface Integrals

13.5.1 Surface Area

13.5.2 Mass and Center of Mass of a Shell

13.5.3 Flux of a Vector Field Across a Surface

13.6 Preparation for the Integral Theorems of Gauss and Stokes

13.7 The Divergence Theorem of Gauss

13.7.1 Archimedes's Principle

13.7.2 The Heat Equation

13.7.3 The Divergence Theorem As A Conservation of Mass Principle

13.7.4 Green's Identities

13.8 The Integral Theorem of Stokes

13.8.1 An Interpretation of Curl

13.8.2 Potential Theory in 3- Space

**Part V - Fourier Analysis, Orthogonal Expansions and Wavelets **

Chapter 14 - Fourier Series

14.1 Why Fourier Series?

14.2 The Fourier Series of a Function

14.2.1 Even and Odd Functions

14.3 Convergence of Fourier Series

14.3.1 Convergence at the End Points

14.3.2 A Second Convergence Theorem

14.3.3 Partial Sums of Fourier Series

14.3.4 The Gibbs Phenomenon

14.4 Fourier Cosine and Sine Series

14.4.1 The Fourier Cosine Series of a Function

14.4.2 The Fourier Sine Series of a Function

14.5 Integration and Differentiation of Fourier Series

14.6 The Phase Angle Form of a Fourier Series

14.7 Complex Fourier Series and the Frequency Spectrum

14.7.1 Review of Complex Numbers

14.7.2 Complex Fourier Series

Chapter 15 - The Fourier Integral and Fourier Transforms

15.1 The Fourier Integral

15.2 Fourier Cosine and Sine Integrals

15.3 The Complex Fourier Integral and the Fourier Transform

15.4 Additional Properties and Applications of the Fourier Transform

15.4.1 The Fourier Transform of a Derivative

15.4.2 Frequency Differentiation

15.4.3 The Fourier Transform of an Integral

15.4.4 Convolution

15.4.5 Filtering and the Dirac Delta Function

15.4.6 The Windowed Fourier Transform

15.4.7 The Shannon Sampling Theorem

15.4.8 Lowpass and Bandpass Filters

15.5 The Fourier Cosine and Sine Transforms

15.6 The Finite Fourier Cosine and Sine Transforms

15.7 The Discrete Fourier Transform

15.7.1 Linearity and Periodicity

15.7.2 The Inverse N- Point DFT

15.7.3 DFT Approximation of Fourier Coefficients

15.8 Sampled Fourier Series

15.8.1 Approximation of a Fourier Transform by an N- Point DFT

15.8.2 Filtering

15.9 The Fast Fourier Transform

15.9.1 Computational Efficiency of the FFT

15.9.2 Use of the FFT in Analyzing Power Spectral Densities of Signals

15.9.3 Filtering Noise From a Signal

15.9.4 Analysis of the Tides in Morro Bay

Chapter 16 - Special Functions, Orthogonal Expansions and Wavelets

16.1 Legendre Polynomials

16.1.1 A Generating Function for the Legendre Polynomials

16.1.2 A Recurrence Relation for the Legendre Polynomials

16.1.3 Orthogonality of the Legendre Polynomials

16.1.4 Fourier-Legendre Series

16.1.5 Computation of Fourier-Legendre Coefficients

16.1.6 Zeros of the Legendre Polynomials

16.1.7 Derivative and Integral Formulas for Pn(x)

16.2 Bessel Functions

16.2.1 The Gamma Function

16.2.2 Bessel Functions of the First Kind and Solutions of Bessel's Equation

16.2.3 Bessel Functions of the Second Kind

16.2.4 Modified Bessel Functions

16.2.5 Some Applications of Bessel Functions

16.2.6 A Generating Function for Jn(x)

16.2.7 An Integral Formula for Jn(x)

16.2.8 A Recurrence Relation for Jv(x)

16.2.9 Zeros of Jv(x)

16.2.10Fourier-Bessel Expansions

16.2.11Fourier-Bessel Coefficients

16.3 Sturm-Liouville Theory and Eigenfunction Expansions

16.3.1 The Sturm-Liouville Problem

16.3.2 The Sturm-Liouville Theorem

16.3.3 Eigenfunction Expansions

16.3.4 Approximation In the Mean and Bessel's Inequality

16.3.5 Convergence in the Mean and Parseval's Theorem

16.3.6 Completeness of the Eigenfunctions

16.4 Orthogonal Polynomials

16.4.1 Chebyshev Polynomials

16.4.2 Laguerre Polynomials

16.4.3 Hermite Polynomials

16.5 Wavelets

16.5.1 The Idea Behind Wavelets

16.5.2 The Haar Wavelets

16.5.3 A Wavelet Expansion

16.5.4 Multiresolution Analysis With Haar Wavelets

16.5.5 General Construction of Wavelets and Multiresolution Analysis

16.5.6 Shannon Wavelets

**Part VI - Partial Differential Equations **

Chapter 17 - The Wave Equation

17.1 The Wave Equation and Initial and Boundary Conditions

17.2 Fourier Series Solutions of the Wave Equation

17.2.1 Vibrating String With Zero Initial Velocity

17.2.2 Vibrating String with Given Initial Velocity and Zero Initial Displacement

17.2.3 Vibrating String With Initial Displacement and Velocity

17.2.4 Verification of Solutions

17.2.5 Transformation of Boundary Value Problems Involving the Wave Equation

17.2.6 Effects of Initial Conditions and Constants on the Motion

17.2.7 Numerical Solution of the Wave Equation

17.3 Wave Motion Along Infinite and Semi-Infinite Strings

17.3.1 Wave Motion Along an Infinite String

17.3.2 Wave Motion Along a Semi-Infinite String

17.3.3 Fourier Transform Solution of Problems on Unbounded Domains

17.4 Characteristics and d'Alembert's Solution

17.4.1 A Nonhomogeneous Wave Equation

17.4.2 Forward and Backward Waves

17.5 Normal Modes of Vibration of a Circular Elastic Membrane

17.6 Vibrations of a Circular Elastic Membrane, Revisited

17.7 Vibrations of a Rectangular Membrane

Chapter 18 - The Heat Equation

18.1 The Heat Equation and Initial and Boundary Conditions

18.2 Fourier Series Solutions of the Heat Equation

18.2.1 Ends of the Bar Kept at Temperature Zero

18.2.2 Temperature in a Bar With Insulated Ends

18.2.3 Temperature Distribution in a Bar With Radiating End

18.2.4 Transformations of Boundary Value Problems Involving the Heat Equation

18.2.5 A Nonhomogeneous Heat Equation

18.2.6 Effects of Boundary Conditions and Constants on Heat Conduction

18.2.7 Numerical Approximation of Solutions

18.3 Heat Conduction in Infinite Media

18.3.1 Heat Conduction in an Infinite Bar

18.3.2 Heat Conduction in a Semi-Infinite Bar

18.3.3 Integral Transform Methods for the Heat Equation in an Infinite Medium

18.4 Heat Conduction in an Infinite Cylinder

18.5 Heat Conduction in a Rectangular Plate

Chapter 19 - The Potential Equation

19.1 Harmonic Functions and the Dirichlet Problem

19.2 Dirichlet Problem for a Rectangle

19.3 Dirichlet Problem for a Disk

19.4 Poisson's Integral Formula for the Disk

19.5 Dirichlet Problems in Unbounded Regions

19.5.1 Dirichlet Problem for the Upper Half Plane

19.5.2 Dirichlet Problem for the Right Quarter Plane

19.5.3 An Electrostatic Potential Problem

19.6 A Dirichlet Problem for a Cube

19.7 The Steady-State Heat Equation for a Solid Sphere

19.8 The Neumann Problem

19.8.1 A Neumann Problem for a Rectangle

19.8.2 A Neumann Problem for a Disk

19.8.3 A Neumann Problem for the Upper Half Plane

**Part VII - Complex Analysis **

Chapter 20 - Geometry and Arithmetic of Complex Numbers

20.1 Complex Numbers

20.1.1 The Complex Plane

20.1.2 Magnitude and Conjugate

20.1.3 Complex Division

20.1.4 Inequalities

20.1.5 Argument and Polar Form of a Complex Number

20.1.6 Ordering

20.1.7 Binomial Expansion of (z + w)n

20.2 Loci and Sets of Points in the Complex Plane

20.2.1 Distance

20.2.2 Circles and Disks

20.2.3 The Equation |z - a| = |z - b|

20.2.4 Other Loci

20.2.5 Interior Points, Boundary Points, and Open and Closed Sets

20.2.6 Limit Points

20.2.7 Complex Sequences

20.2.8 Subsequences

20.2.9 Compactness and the Bolzano-Weierstrass Theorem

Chapter 21 - Complex Functions

21.1 Limits, Continuity and Derivatives

21.1.1 Limits

21.1.2 Continuity

21.1.3 The Derivative of a Complex Function

21.1.4 The Cauchy-Riemann Equations

21.2 Power Series

21.2.1 Series of Complex Numbers

21.2.2 Power Series

21.3 The Exponential and Trigonometric Functions

21.4 The Complex Logarithm

21.5 Powers

21.5.1 Integer Powers

21.5.2 z1/n for Positive Integer n

21.5.3 Rational Powers

21.5.4 Powers zw

Chapter 22 - Complex Integration

22.1 Curves in the Plane

22.2 The Integral of a Complex Function

22.2.1 The Complex Integral in Terms of Real Integrals

22.2.2 Properties of Complex Integrals

22.2.3 Integrals of Series of Functions

22.3 Cauchy's Theorem

22.3.1 Proof of Cauchy's Theorem for a Special Case

22.3.2 Proof of Cauchy's Theorem for a Rectangle

22.4 Consequences of Cauchy's Theorem

22.4.1 Independence of Path

22.4.2 The Deformation Theorem

22.4.3 Cauchy's Integral Formula

22.4.4 Cauchy's Integral Formula for Higher Derivatives

22.4.5 Bounds on Derivatives and Liouville's Theorem

22.4.6 An Extended Deformation Theorem

Chapter 23 - Series Representations of Functions

23.1 Power Series Representations

23.1.1 Isolated Zeros and the Identity Theorem

23.1.2 The Maximum Modulus Theorem

23.2 The Laurent Expansion

Chapter 24 - Singularities and the Residue Theorem

24.1 Singularities

24.2 The Residue Theorem

24.3 Some Applications of the Residue Theorem

24.3.1 The Argument Principle

24.3.2 Rouché's Theorem

24.3.3 Summation of Real Series

24.3.4 An Inversion Formula for the Laplace Transform

24.3.5 Evaluation of Real Integrals

Chapter 25 - Conformal Mappings

25.1 Functions as Mappings

25.2 Conformal Mappings

25.2.1 Linear Fractional Transformations

25.3 Construction of Conformal Mappings Between Domains

25.3.1 Schwarz-Christoffel Transformation

25.4 Harmonic Functions and the Dirichlet Problem

25.4.1 Solution of Dirichlet Problems by Conformal Mapping

25.5 Complex Function Models of Plane Fluid Flow

**Part VIII - Probability and Statistics **

Chapter 26 - Counting and Probability

26.1 The Multiplication Principle

26.2 Permutations

26.3 Choosing r Objects From n Objects

26.3.1 r Objects From n Objects, With Order

26.3.2 r Objects From n Objects, Without Order

26.3.3 Tree Diagrams

26.4 Events and Sample Spaces

26.5 The Probability of an Event

26.6 Complementary Events

26.7 Conditional Probability

26.8 Independent Events

26.8.1 The Product Rule

26.9 Tree Diagrams In Computing Probabilities

26.10 Bayes' Theorem

26.11 Expected Value

Chapter 27 - Statistics

27.1 Measures of Center and Variation

27.1.1 Measures of Center

27.1.2 Measures of Variation

27.2 Random Variables and Probability Distributions

27.3 The Binomial and Poisson Distributions

27.3.1 The Binomial Distribution

27.3.2 The Poisson Distribution

27.4 A Coin Tossing Experiment, Normally Distributed Data, and the Bell Curve

27.4.1 The Standard Bell Curve

27.4.2 The 68, 95, 99.7 Rule

27.5 Sampling Distributions and the Central Limit Theorem

27.6 Confidence Intervals and Estimating Population Proportion

27.7 Estimating Population Mean and the Student t Distribution

27.8 Correlation and Regression

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