Projects for Section 2.2:

When Differential Equations Invaded Geometry: Inverse Tangent Problems in the 17th Century

Projects for Section 2.5:

Two properties of the sphere

Projects for Section 2.7:

PotassiumArgon Dating

Projects for Section 2.8:

Tricky timing: The Isochrones of Huygenes and Leibniz

Projects for Section 3.8:

Vibration control:Vibration Isolation

Projects for Section 3.11:

Vibration control:Vibration Absorbers

Projects for Section 9.16:

Minimal Surfaces

Projects for Section 13.4:

Making waves: Convection, Diffusion, and Traffic Flow

Projects for Section 14.4:

The uncertainity inequality in signal processing


Part 1: Ordinary Differential Equations

Chapter 1: Introduction to Differential Equations

Definitions and Terminology

InitialValue Problems

Differential Equations as Mathematical Models

Chapter 1 in Review


Chapter 2: FirstOrder Differential Equations

Solution Curves Without a Solution: Direction Fields

Autonomous FirstOrder DEs

Separable Equations

Linear Equations

Exact Equations

Solutions by Substitutions

A Numerical Method

Linear Models

Nonlinear Models

Modeling with Systems of FirstOrder DEs

Chapter 2 in Review


Chapter 3: HigherOrder Differential Equations

Theory of Linear Equations: InitialValue and BoundaryValue Problems

Homogeneous Equations

Non Homogeneous Equations

Reduction of Order

Homogenous Liner Equations with Constant Coefficients

Undetermined Coefficients

Variation of Parameters

CauchyEuler Equation

Nonlinear Equations

Linear Models: InitialValue Problem:Spring/Mass Systems: Free Undamped Motion

Spring/Mass Systems: Free Damped Motion

Spring/Mass System: Driven Motion

Series Circuit Analogue

Linear Models: BoundaryValue Problems

Greens Functions: InitialValue Problems

BoundaryValue Problems

Nonlinear Models

Solving Systems of Linear Equations

Chapter 3 in Review


Chapter 4: The Laplace Transform

Definition of the Laplace Transform

The Inverse Transform and Transform ofDerivatives:Inverse Transforms

Transforms of Derivatives

Translation Theorems: Translationonthe saxis

Translation on the taxis

Additional Operational Properties: Derivatives of Transforms

Transforms of Integrals

Transform of a Periodic Function

The DiracDelta Function

Systems of LinearDifferential Equations

Chapter 4 in Review


Chapter 5: Series Solutions of Linear Differential Equations

Solutionsabout Ordinary Points: Review of Power Series

Power Series Solutions

Solutions about Singular Points

Special Functions: Bessel Functions

Legendre Functions

Chapter 5 in Review


Chapter 6: Numerical Solutions of Ordinary Differential Equations

Euler Methods and Error Analysis

RungeKutta Methods

Multistep Methods

HigherOrder Equations and Systems

SecondOrder BoundaryValue Problems

Chapter 6 in Review


Part 2: Vectors, Matrices, and Vector Calculus


Chapter 7: Vectors

Vectors in 2Space

Vectors in 3Space

Dot Product

Cross Product

Lines and Planes in 3Space

Vector Spaces

GramSchmidt Orthogonalization Process

Chapter 7 in Review


Chapter 8: Matrices

Matrix Algebra

Systems of Linear Algebraic Equations

Rank of a Matrix

Determinants

Properties of Determinants

Inverse of a Matrix: Finding the Inverse

Using the Inverse to Solve Systems

Cramers Rule

The Eigenvalue Problem

Powers of Matrices

Orthogonal Matrices

Approximation of Eigenvalues

Diagonalization

Cryptography

An ErrorCorrecting Code

Method of Least Squares

Discrete Compartmental Models

Chapter 8 in Review


Chapter 9: Vector Calculus

Vector Functions

Motion on a Curve

Curvature and Components of Acceleration

Partial Derivatives

Directional Derivative

Tangent Planes and Normal Lines

Curl and Divergence

Line Integrals

Independence of the Path

Double Integrals

Double Integrals in Polar Coordinates

Greens Theorem

Surface Integrals

Stokes Theorem

Triple Integrals

Divergence Theorem

Change of Variables in Multiple Integrals

Chapter 9 in Review


Part 3: Systems of Differential Equations

Chapter 10: Systems of Linear Differential Equations

Theory of Linear Systems

Homogeneous Linear Systems: Distinct Real Eigenvalues

Repeated Eigenvalues

Complex Eigenvalues

Solution by Diagonalization

NonhomogeneousLinear Systems: Undetermined Coefficients

Variation of Parameters

Diagonalization

Matrix Exponential

Chapter 10 in Review


Chapter 11: Systems of Nonlinear Differential Equations

Autonomous Systems

Stability of Linear Systems

Linearization and Local Stability

Autonomous Systems as Mathematical Models

Periodic Solutions, Limit Cycles, and Global Stability

Chapter 11 in Reviews


Part 4: Partial Differential Equations


Chapter 12: Orthogonal Functions and Fourier Series

Orthogonal Functions

Fourier Series

Fourier Cosine and Sine Series

Complex Fourier Series

StrumLiouville Problem

Bessel and Legendre Series: FourierBessel Series

FourierLegendre Series

Chapter 12 in Review


Chapter 13: BoundaryValue Problems in Rectangular Coordinates

Separable Partial Differential Equations

Classical PDEs and BoundaryValue Problems

Heat Equation

Wave Equation

Laplaces Equation

Nonhomogeneous BVPs

Orthogonal Series Expansions

Fourier Series in Two Variables

Chapter 13 in Review


Chapter 14: BoundaryValue Problems in Other Coordinate Systems

Problems in Polar Coordinates

Problems in Cylindrical Coordinates

Problems in Spherical Coordinates

Chapter 14 in Review


Chapter 15: Integral Transform Method

Error Function

Applications of the Laplace Transform

Fourier Integral

Fourier Transforms

Fast Fourier Transform

Chapter 15 in Review


Chapter 16: Numerical Solutions of Partial Differential Equations

Laplaces Equation

The Heat Equation

The Wave Equation

Chapter 16 in Review


Part 5: Complex Analysis


Chapter 17: Functions of a Complex Variable

Complex Numbers

Powers and Roots

Sets in the Complex Plane

Functions of a Complex Variable

CauchyRiemann Equations

Exponential and Logarithmic Functions

Trigonometric and Hyperbolic Functions

Inverse Trigonometric and Hyperbolic Functions

Chapter 17 in Review


Chapter 18: Integration in the Complex Plane

Contour Integrals

CauchyGoursat Theorem

Independence of Path

Cauchys Integral Formulas

Chapter 18 in Review


Chapter 19: Series and Residues

Sequences and Series

Taylor Series

Laurent Series

Zeros and Poles

Residues and Residue Theorem

Evaluation of Real Integrals

Chapter 19 in Review


Chapter 20: Conformal Mappings

ComplexFunctions as Mappings

Conformal Mappings

Linear Fractional Transformations

SchwarzChristoffel Transformations

Poisson Integral Formulas

Applications


Chapter 20 in Review

Appendix 1: Derivative and Integral Formula

Appendix II: Gamma Function

Appendix III: Table of Laplace Transforms

Appendix IV: Conformal Mappings

Answers for Selected Oddnumbered Problems

Index
