Advanced Engeneering Mathematics

In this text, results are typically stated “up front,” either informally, or formally as a theorem, then illustrated with examples before being proved or verified. A conscious effort has been made to ensure that students will understand what a theorem is saying, before they are subjected to its proof. While this is not the standard ordering one finds in math texts in general, it is the ordering often found in classrooms where applications rule.

Nearly 7000 exercises are available for student practice and enrichment. Nearly all are completely solved in the Instructor's Manual.

This text is the outgrowth of more than ten years of using Maple in the classroom to teach science and engineering students courses in calculus, differential equations, linear algebra, boundary value problems, advanced calculus, vector calculus, complex variables, and statistics. The materials were conceived in laboratory/classrooms where each student sat at a desktop machine, and nurtured in an environment where each student carries a laptop computer the way we old-timers carried our slide rules. Over the years, the materials were presented at various stages in journal articles, conference talks, workshops, and seminars.

Throughout, the theme guiding their development has been the realization that modern computer algebra systems and software pose a new paradigm for teaching, learning, and doing applied mathematics. Indeed, the phrase “new apprenticeship” echoes in the writings and talks leading up to this present volume. It is no longer enough to acknowledge the power of such software while still adhering to the paradigm of pencil-and-paper, and the chalkboard.

Paralleling the text, therefore, is a collection of Maple worksheets, which implement all the calculations and derivations found described in this book. In fact, each of the book's 273 sections is mirrored in a worksheet that includes both the prose and the mathematics, the mathematics being “live” in the worksheet. Students reading the text can have the parallel discussion on their computer screens, and can execute in Maple, the calculations the text is describing.

Yet, it is entirely possible to “lecture” from this text. There is ample opportunity for an instructor to reproduce calculations and derivations summarized in these pages. While such lectures are being delivered, it is hoped that students will interact with the material by using a computer algebra system to interpret the mathematics, and to work the exercises.

Two-thirds of the exercise sets are divided into problems of types A and B. Problems of type A are more conceptual, and less demanding computationally. The A-problems would be the ones a student might work by hand if they found that to be an effective way to learn. The B-problems are generally more computationally intensive. It is anticipated that technological tools of some sort will be used freely when working these exercises.

Not all in the math, science, and engineering communities have embraced the use of technology as the operational instrument for meeting, and mastering, mathematics. Many times, both on my own campus and on others, I have had to articulate the case for the active use of technology as the learning agent in math courses. Typically, I would try to show by examples how technology has improved pedagogy. Sometimes, I would say something like “The course of instruction at a school for operators of earth-moving machinery should not end in a test of dexterity with a shovel, nor should admission be limited to those capable of digging a ditch with one.” But my favorite analogy is that of the Magic Skates, born of my experiences at the ice-hockey rinks in Canada where I lived for twelve years.

If you can't skate, you can't play hockey. Only youngsters who master the art of skating can experience the game of hockey. But suppose a poor skater acquires a pair of magic skates which transform the wearer into an adept skater capable of experiencing the thrill of the game of hockey. Is it viable to argue that the magic skates invalidate the player's ensuing encounter with the game?

This text embraces the magic skates of computer algebra systems. Every volunteer hockey coach I knew back in Canada would have paid for pairs of magic skates from their own pockets, just to see their teams play a better game of hockey. It was the game that mattered, the play, the experience, the participation in a really exciting sport. And if we can't make our students feel the same way about applied and engineering mathematics, our programs will retain only the dwindling handful willing to make the 5:30 AM practice before school.

Distinctive Features
1. New Paradigm
Access to computer algebra tools can be assumed throughout, and the pedagogy can be predicated on its availability and use. Although the text is written in traditional notation, its structure reflects the author's experience in using a computer as an active partner in teaching, learning, and doing applied and engineering mathematics.

2. New Apprenticeship
Insights into the deep results of classical applied mathematics are extracted from examples, as much by calculations and graphics as by subtle reasoning. This text shows how to use modern software tools to learn, do, and interpret applied and engineering mathematics.

3. Flexibility
A computer algebra system allows the instructor the option to bypass certain drills in skill-building, to concentrate on key ideas. Therefore, topics can be reordered more easily whenever supporting computations can be relegated to the computer.

4. “Big Picture” First
Reflecting the author's own learning style, most presentations begin with the “big picture,” with computations and supporting graphics given first. Then, when the goal is clear, the supporting calculations and relationships are developed.

5. Parallel Worksheets
The 273 sections of the text are paralleled by a matching number of Maple worksheets containing, not only the calculations and graphics of the section, but also the text and explanations. The student using the Maple worksheet sees more than just a computer dialog. The complete text is included in the worksheets, with detailed explanations of both the mathematics and the Maple commands required to obtain it.

6. Pervasive Access to Mathematical Tools
Relying on a computer algebra system allows mathematical tools to be used before they are developed formally in the text. For example, numerical evaluation of integrals occurs well before the formal treatment of numeric integration in Unit Eight. Eigenvalues are computed numerically in Unit Three, before the chapters on numerical methods.

7. Complete Integration of Numeric and Symbolic Results
Numeric results are interwoven with symbolic calculations throughout the text. Numeric solutions for differential equations appear early enough to be used throughout the study of models based on differential equations. The perturbation techniques of Poincare, Lindstedt, and Krylov-Bogoliubov are in the same unit as the second-order IVP. Collocation, Rayleigh-Ritz, and Galerkin techniques for solving BVPs are contiguous with analytic techniques, and with finite-difference, finite-element, and shooting techniques. Later, in Unit Five, numeric methods for solving PDEs also appear in conjunction with the more classical symbolic results.

8. Early Appearance of the Laplace Transform
The Laplace transform as a tool for solving IVPs for ordinary differential equations appears in Chapter 6. This makes it available in Unit Three where systems of ODEs are studied.

9. Integration of Matrix Algebra with Systems of ODEs
Systems of first-order linear ODEs motivate and drive vector and matrix manipulations. Chemical mixing tanks provide the model, the Laplace transform is used to obtain solutions, and the vector-matrix structure in the model and its solution is deduced. This motivates a study of the eigenvalue problem, and leads to the fundamental matrix, first via the Laplace transform, then as the exponential of a matrix. Necessary matrix algebra is developed in the context of linear systems of ODEs.

10. Two Types of Exercises, Part A and Part B
The exercises in approximately two-thirds of the sections are divided into two categories. The A-exercises are generally more conceptual, and can be done without a suite of computer tools. The B-exercises generally presuppose access to appropriate computer tools, and provide both practice for the section and generalizations beyond the text.

11. A Unit on Series
A unit discussing power series, Fourier series, and asymptotic series sits between the two units on ordinary differential equations. Solutions represented in these forms then appear in Unit Three, the second unit on ODEs.

12. A Unit on the Calculus of Variations
A unit on the Calculus of Variations (Unit Nine) is available as a supplement.

13. Socratic Chapter Reviews
The many questions (rather than new exercises) in a Chapter Review aid the student in organizing the chapter's material.

Supplements
Unit 9: Calculus of Variations
(0-201-72204-6)
This unit includes the chapters “Basic Formalisms,” “Constrained Optimization,” and “Variational Mechanics.”

Instructor's Technology Resource & Solutions Manual
(0-201-71001-3)
This manual includes:

• Introduction to & Tips for Maple^®
• Introduction to & Tips for Mathematica^®
• Solutions to A Exercises
• A CD-ROM in the back of the manual includes:
  • Fully worked solutions to B exercises in Maple^® Worksheets
  • Fully worked solutions to B exercises in Mathematica^® Notebooks
  • Free Mathematica^® Reader

• Introduction to & Tips for Maple^®
• Introduction to & Tips for Mathematica^®
• Solutions to Selected A Answers
• A CD-ROM in the back of the manual includes:
  • Fully worked selected solutions to B exercises in Maple^® Worksheets
  • Fully worked selected solutions to B exercises in Mathematica^® Notebooks
  • Free Mathematica^® Reader

Contents

Preface

UNIT I. Ordinary Differential Equations-Part One

Chapter 1 First-Order Differential Equations

1.1 Introduction

1.2 Terminology

1.3 The Direction Field

1.4 Picard Iteration

1.5 Existence and Uniqueness for the Initial Value Problem

Chapter 2 Models Containing ODEs

2.1 Exponential Growth and Decay

2.2 Logistic Models

2.3 Mixing Tank Problems-Constant and Variable Volumes

2.4 Newton's Law of Cooling

Chapter 3 Methods for Solving First-Order ODEs

3.1 Separation of Variables

3.2 Equations with Homogeneous Coefficients

3.3 Exact Equations

3.4 Integrating Factors and the First-Order Equations

3.5 Variation of Parameters and the First-Order Linear Equation

3.6 The Bernoulli Equation

Chapter 4 Numeric Methods for Solving First-Order ODEs

4.1 Fixed-Step methods-Order and Error

4.2 The Euler Method

4.3 Taylor Series Methods

4.4 Runge-Kutta Methods

4.5 Adams-Bashforth Multistep Methods

4.6 Adams-Moulton Predictor-Corrector Methods

4.7 Milne's Method

4.8 rkf45, the Runge-Kutta-Fehlberg Method

Chapter 5 Second-Order Differential Equations

5.1 Springs 'n' Things

5.2 The Initial Value Problem

5.3 Overview of Solution Process

5.4 Linear Dependence and Independence

5.5 Free Undamped Motion

5.6 Free Damped Motion

5.7 Reduction of Order and Higher-Order Equations

5.8 The Bobbing Cylinder

5.9 Forced Motion and Variation of Parameters

5.10 Forced Motion and Undetermined Coefficients

5.11 Resonance

5.12 The Euler Equation

5.13 The Green's Function Technique for IVPS

Chapter 6 The Laplace Transform

6.1 Definition and Examples

6.2 Transform of Derivatives

6.3 First Shifting Law

6.4 Operational Laws

6.5 Heaviside Functions and the Second Shifting Law

6.6 Pulses and the Third Shifting Law

6.7 Transforms of Periodic Functions

6.8 Convolution and the Convolution Theorem

6.9 Convolution Products by the Convolution Theorem

6.10 The Dirac Delta Function

6.11 Transfer Function, Fundamental Solution, and the Green's Function

UNIT II. Infinite Series

Chapter 7 Sequences and Series of Numbers

7.1 Sequences

7.2 Infinite Series

7.3 Series with Positive Terms

7.4 Series with Both Negative and Positive Terms

Chapter 8 Sequences and Series of Functions

8.1 Sequences of Functions

8.2 Pointwise Convergence

8.3 Uniform Convergence

8.4 Convergence in the Mean

8.5 Series of Functions

Chapter 9 Power Series

9.1 Taylor Polynomials

9.2 Taylor Series

9.3 Termwise Operations on Taylor Series

Chapter 10 Fourier Series

10.1 General Formalism

10.2 Termwise Integration and Differentiation

10.3 Odd and Even Functions and their Fourier Series

10.4 Sine Series and Cosine Series

10.5 Periodically Driven Damped Oscillator

10.6 Optimizing Property of Fourier Series

10.7 Fourier-Legendre Series

Chapter 11 Asymptotic Series

11.1 Computing with Divergent Series

11.2 Definitions

11.3 Operations with Asymptotic Series

UNIT III. Ordinary Differential Equations-Part Two

Chapter 12 Systems of First-Order ODEs

12.1 Mixing tanks-Closed Systems

12.2 Mixing tanks-Open Systems

12.3 Vector Structure of Solutions

12.4 Determinants and Cramer's Rule

12.5 Solving Linear Algebraic Equations

12.6 Homogeneous Equations and the Null Space

12.7 Inverses

12.8 Vectors and the Laplace Transform

12.9 The Matrix Exponential

12.10 Eigenvalues and Eigenvectors

12.11 Solutions by Eigenvalues and Eigenvectors

12.12 Finding Eigenvalues and Eigenvectors

12.13 System versus. Second-Order ODE

12.14 Complex Eigenvalues

12.15 The Deficient Case

12.16 Diagonalization and Uncoupling

12.17 A Coupled Linear Oscillator

12.18 Nonhomogeneous Systems and Variation of Parameters

12.19 Phase Portraits

12.20 Stability

12.21 Nonlinear Systems

12.22 Linearization

12.23 The Nonlinear Pendulum

Chapter 13 Numerical Techniques: First-Order Systems and Second-Order ODEs

13.1 Runge-Kutta-Nystrom

13.2 rk4 for First-Order Systems

Chapter 14 Series Solutions

14.1 Power series

14.2 Asymptotic solutions

14.3 Perturbation Solution of an Algebraic Equation

14.4 Poincaré Perturbation Solution for Differential Equations

14.5 The Nonlinear Spring and Lindstedt's Method

14.6 The Method of Krylov and Bogoliubov

Chapter 15 Boundary Value Problems

15.1 Analytic Solutions

15.2 Numeric Solutions

15.3 Least-squares, Rayleigh-Ritz, Galerkin, and Collocation Techniques

15.4 Finite Elements

Chapter 16 The Eigenvalue Problem

16.1 Regular Sturm-Liouville Problems

16.2 Bessel's Equation

16.3 Legendre's Equation

16.4 Solution by Finite Differences

UNIT IV. Vector Calculus

Chapter 17 Space Curves

17.1 Curves and Their Tangent Vectors

17.2 Arc Length

17.3 Curvature

17.4 Principal Normal and Binormal Vectors

17.5 Resolution of R² into Tangential and Normal Components

17.6 Applications to Dynamics

Chapter 18 The Gradient Vector

18.1 Visualizing Vector Fields and Their Flows

18.2 The Directional Derivative and Gradient Vector

18.3 Properties of the Gradient Vector

18.4 Lagrange Multipliers

18.5 Conservative Forces and the Scalar Potential

Chapter 19 Line Integrals in the Plane

19.1 Work and Circulation

19.2 Flux Through a Plane Curve

Chapter 20 Additional Vector Differential Operators

20.1 Divergence and Its Meaning

20.2 Curl and Its Meaning

20.3 Products-One Del and Two Operands

20.4 Products-Two Dels and One Operand

>Chapter 21 Integration

21.1 Surface Area

21.2 Surface Integrals and Surface Flux

21.3 The Divergence Theorem and the Theorems of Green and Stokes

21.4 Green's Theorem

21.5 Conservative, Solenoidal, and Irrotational Fields

21.6 Integral Equivalents of div, grad, and curl

Chapter 22 Non-Cartesian Coordinates

22.1 Mappings and Changes of Coordinates

22.2 Vector Operators in Polar Coordinates

22.3 Vector Operators in Cylindrical and Spherical Csoordinates

Chapter 23 Miscellaneous Results

23.1 Gauss' Theorem

23.2 Surface Area for Parametrically Given Surfaces

23.3 The Equation of Continuity

23.4 Green's Identities

UNIT V. Boundary Value Problems for PDEs

Chapter 24 Wave Equation

24.1 The Plucked String

24.2 The Struck String

24.3 D'Alembert's Solution

24.4 Derivation of the Wave Equation

24.5 Longitudinal Vibrations in an Elastic Rod

24.6 Finite-Difference Solution of the One-Dimensional Wave Equation

Chapter 25 Heat Equation

25.1 One-Dimensional Heat Diffusion

25.2 Derivation of the One-Dimensional Heat Equation

25.3 Heat Flow in a Rod with Insulated Ends

25.4 Finite-Difference Solution of the One-Dimensional Heat Equation

Chapter 26 Laplace's Equation on a Rectangle

26.1 Nonzero Temperature on the Bottom Edge

26.2 Nonzero Temperature on the Top Edge

26.3 Nonzero Temperature on the Left Edge

26.4 Finite-Difference Solution of Laplace's Equation on a Rectangle

Chapter 27 Nonhomogeneous Boundary Value Problems

27.1 One-Dimensional Heat Equation with Different Endpoint Temperatures

27.2 One-Dimensional Heat Equation with Time-Varying Endpoint Temperatures

Chapter 28 Time-Dependent Problems in Two Spatial Dimensions

28.1 Oscillations of a Rectangular Membrane

28.2 Time-Varying Temperatures on a Rectangular Plate

Chapter 29 Separation of Variables in Non-Cartesian Coordinates

29.1 Laplace's Equation in a Disk

29.2 Laplace's Equation in a Cylinder

29.3 The Circular Drumhead

29.4 Laplace's Equation in a Sphere

29.5 The Spherical Dielectric

Chapter 30 Transform Techniques

30.1 Solution by Laplace Transform

30.2 The Fourier Integral Theorem

30.3 The Fourier Transform

30.4 Wave Equation on the Infinite String-Solution by Fourier Transform

30.5 Heat Equation on the Infinite Rod-Solution by Fourier Transform

30.6 Laplace's Equation on the Infinite Strip-Solution by Fourier Transform

30.7 The Fourier Sine Transform

30.8 The Fourier Cosine Transform

UNIT VI. Matrix Algebra

Chapter 31 Vectors as Arrows

31.1 The Algebra and Geometry of Vectors

31.2 Inner and Dot Products

31.3 The Cross-Product

Chapter 32 Change of Coordinates

32.1 Change of Basis

32.2 Rotations and Orthogonal Matrices

32.3 Change of Coordinates

32.4 Reciprocal Bases and Gradient Vectors

32.5 Gradient Vectors and the Covariant Transformation Law

Chapter 33 Matrix Computations

33.1 Summary

33.2 Projections

33.3 The Gram-Schmidt Orthogonalization Process

33.4 Quadratic Forms

33.5 Vector and Matrix Norms

33.6 Least Squares

Chapter 34 Matrix Factorizations

34.1 LU Decomposition

34.2 PJP^-1 and Jordan Canonical Form

34.3 QR Decomposition

34.4 QR Algorithm for Finding Eigenvalues

34.5 SVD, The Singular Value Decomposition

34.6 Minimum-Length Least-Squares Solution, and the Pseudoinverse

UNIT VII. Complex Variables

Chapter 35 Fundamentals

35.1 Complex Numbers

35.2 The Function w = f(z) = z²

35.3 The Function w = f(z) = z³

35.4 The Exponential Function

35.5 The Complex Logarithm

35.6 Complex Exponents

35.7 Trigonometric and Hyperbolic Functions

35.8 Inverses of Trigonometric and Hyperbolic Functions

35.9 Differentiation and the Cauchy-Riemann Equations

35.10 Analytic and Harmonic Functions

35.11 Integration

35.12 Series in Powers of z

35.13 The Calculus of Residues

Chapter 36 Applications

36.1 Evaluation of Integrals

36.2 The Laplace Transform

36.3 Fourier Series and the Fourier Transform

36.4 The Root Locus

36.5 The Nyquist Stability Criterion

36.6 Conformal Mapping

36.7 The Joukowski Map

36.8 Solving the Dirichlet Problem by Conformal Mapping

36.9 Planar Fluid Flow

36.10 Conformal Mapping of Elementary Flows

UNIT VIII. Numerical Methods

Chapter 37 Equations in One Variable-Preliminaries

37.1 Accuracy and Errors

37.2 Rate of Convergence

Chapter 38 Equations in One Variable-Methods

38.1 Fixed-Point Iteration

38.2 The Bisection Method

38.3 Newton-Raphson Iteration

38.4 The Secant Method

38.5 Muller's Method

Chapter 39 Systems of Equations

39.1 Gaussian Arithmetic

39.2 Condition Numbers

39.3 Iterative Improvement

39.4 The Method of Jacobi

39.5 Gauss-Seidel Iteration

39.6 Relaxation and SOR

39.7 Iterative Mmethods for Nonlinear Systems

39.8 Newton's Iteration for Nonlinear Systems

Chapter 40 Interpolation

40.1 Lagrange Interpolation

40.2 Divided Differences

40.3 Chebyshev Interpolation

40.4 Spline Interpolation

40.5 Bezier Curves

Chapter 41 Approximation of Continuous Functions

41.1 Least-Squares Approximation

41.2 Padé Approximations

41.3 Chebyshev Approximation

41.4 Chebyshev-Padé and Minimax Approximations

Chapter 42 Numeric Differentiation

42.1 Basic Formulas

42.2 Richardson Extrapolation

Chapter 43 Numeric Integration

43.1 Methods from Elementary Calculus

43.2 Recursive Trapezoid rule and Romberg Integration

43.3 Gauss-Legendre Quadrature

43.4 Adaptive Quadrature

43.5 Iterated Integrals

Chapter 44 Approximation of Discrete Data

44.1 Least-Squares Regression Line

44.2 The General Linear Model

44.3 The Role of Orthogonality

44.4 Nonlinear Least Squares

Chapter 45 Numerical Calculation of Eigenvalues

45.1 Power Methods

45.2 Householder Reflections

45.3 QR Decomposition via Householder Reflections

45.4 Upper-Hessenberg Form, Givens Rotations, and the Shifted QR-Algorithm

45.5 The Generalized Eigenvalue Problem

UNIT IX. Calculus of Variations

Chapter 46 Basic Formalisms

46.1 Motivational Examples

46.2 Direct Methods

46.3 The Euler-Lagrange Equation

46.4 First Integrals

46.5 Derivation of the Euler-Lagrange Equation

46.6 Transversality Conditions

46.7 Derivation of the Transversality Conditions

46.8 Three generalizations

Chapter 47 Constrained Optimization

47.1 Applications of Lagrange Multipliers

47.2 Queen Dido's Problem

47.3 Isoperimetric Problems

47.4 The Hanging Chain

47.5 A Variable-Endpoint Problem

47.6 Differential Constraints

Chapter 48 Variational Mechanics

48.1 Hamilton's Principle

48.2 The Simple Pendulum

48.3 A Compound Pendulum

48.4 The Spherical Pendulum

48.5 Pendulum with Oscillating Support

48.6 Legendre and Extended Legendre Transformations

48.7 Hamilton's Canonical Equations

Answers to Selected Exercises

Bibliography

Index