Advanced engineering mathematics, Lecture notes of Mathematics

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Alan Jeffrey

University of Newcastle-upon-Tyne

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C O N T E N T S

Preface xv

P A R T O N E REVIEW MATERIAL 1

C H A P T E R 1 Review of Prerequisites 3 1.1 Real Numbers, Mathematical Induction, and Mathematical Conventions 4 1.2 Complex Numbers 10 1.3 The Complex Plane 15 1.4 Modulus and Argument Representation of Complex Numbers 18 1.5 Roots of Complex Numbers 22 1.6 Partial Fractions 27 1.7 Fundamentals of Determinants 31 1.8 Continuity in One or More Variables 35 1.9 Differentiability of Functions of One or More Variables 38 1.10 Tangent Line and Tangent Plane Approximations to Functions 40 1.11 Integrals 41 1.12 Taylor and Maclaurin Theorems 43 1.13 Cylindrical and Spherical Polar Coordinates and Change of Variables in Partial Differentiation 46 1.14 Inverse Functions and the Inverse Function Theorem 49

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P A R T T H R E E ORDINARY DIFFERENTIAL EQUATIONS 225

C H A P T E R 5 First Order Differential Equations 227 5.1 Background to Ordinary Differential Equations 228 5.2 Some Problems Leading to Ordinary Differential Equations 233 5.3 Direction Fields 240 5.4 Separable Equations 242 5.5 Homogeneous Equations 247 5.6 Exact Equations 250 5.7 Linear First Order Equations 253 5.8 The Bernoulli Equation 259 5.9 The Riccati Equation 262 5.10 Existence and Uniqueness of Solutions 264

C H A P T E R 6 Second and Higher Order Linear Differential

Equations and Systems 269

6.1 Homogeneous Linear Constant Coefficient Second Order Equations 270 6.2 Oscillatory Solutions 280 6.3 Homogeneous Linear Higher Order Constant Coefficient Equations 291 6.4 Undetermined Coefficients: Particular Integrals 302 6.5 Cauchy–Euler Equation 309 6.6 Variation of Parameters and the Green’s Function 311 6.7 Finding a Second Linearly Independent Solution from a Known Solution: The Reduction of Order Method 321 6.8 Reduction to the Standard Form u ′′^ + f ( x ) u = 0 324 6.9 Systems of Ordinary Differential Equations: An Introduction 326 6.10 A Matrix Approach to Linear Systems of Differential Equations 333 6.11 Nonhomogeneous Systems 338 6.12 Autonomous Systems of Equations 351

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7.3 Systems of Equations and Applications of the

18.6 Classification and Reduction to Standard Form

• P A R T T W O VECTORS AND MATRICES
  • C H A P T E R 2 Vectors and Vector Spaces - 2.1 Vectors, Geometry, and Algebra - 2.2 The Dot Product (Scalar Product) - 2.3 The Cross Product (Vector Product) - and Triple Products - 2.5 n -Vectors and the Vector Space R n - 2.6 Linear Independence, Basis, and Dimension - 2.7 Gram–Schmidt Orthogonalization Process
  • C H A P T E R 3 Matrices and Systems of Linear Equations - 3.1 Matrices - 3.2 Some Problems That Give Rise to Matrices - 3.3 Determinants - and Their Connection with Matrix Multiplication 3.4 Elementary Row Operations, Elementary Matrices, - Forms of a Matrix 3.5 The Echelon and Row-Reduced Echelon - 3.6 Row and Column Spaces and Rank - of Linear Equations 3.7 The Solution of Homogeneous Systems - of Linear Equations 3.8 The Solution of Nonhomogeneous Systems - 3.9 The Inverse Matrix
    • 3.10 Derivative of a Matrix
  • C H A P T E R 4 Eigenvalues, Eigenvectors, and Diagonalization - and Eigenvectors 4.1 Characteristic Polynomial, Eigenvalues, - 4.2 Diagonalization of Matrices - 4.3 Special Matrices with Complex Elements - 4.4 Quadratic Forms - 4.5 The Matrix Exponential
• C H A P T E R 7 The Laplace Transform - 7.1 Laplace Transform: Fundamental Ideas - 7.2 Operational Properties of the Laplace Transform - Laplace Transform - 7.4 The Transfer Function, Control Systems, and Time Lags - Functions, and Sturm–Liouville Equations C H A P T E R 8 Series Solutions of Differential Equations, Special - of Differential Equations 8.1 A First Approach to Power Series Solutions - of Homogeneous Equations 8.2 A General Approach to Power Series Solutions - 8.3 Singular Points of Linear Differential Equations - 8.4 The Frobenius Method - 8.5 The Gamma Function Revisited - 8.6 Bessel Function of the First Kind J n ( x ) - 8.7 Bessel Functions of the Second Kind Y ν ( x ) - 8.8 Modified Bessel Functions I ν ( x ) and K ν ( x ) - 8.9 A Critical Bending Problem: Is There a Tallest Flagpole? - and Orthogonality 8.10 Sturm–Liouville Problems, Eigenfunctions,
  • 8.11 Eigenfunction Expansions and Completeness - THE FOURIER TRANSFORM P A R T F O U R FOURIER SERIES, INTEGRALS, AND
• C H A P T E R 9 Fourier Series - 9.1 Introduction to Fourier Series - and Differentiation 9.2 Convergence of Fourier Series and Their Integration - 9.3 Fourier Sine and Cosine Series on 0 ≤ x ≤ L - 9.4 Other Forms of Fourier Series - 9.5 Frequency and Amplitude Spectra of a Function - 9.6 Double Fourier Series
• C H A P T E R 10 Fourier Integrals and the Fourier Transform - 10.1 The Fourier Integral - 10.2 The Fourier Transform - 10.3 Fourier Cosine and Sine Transforms
  • P A R T F I V E VECTOR CALCULUS
• C H A P T E R 11 Vector Differential Calculus - and Differentiability 11.1 Scalar and Vector Fields, Limits, Continuity, - of a Single Real Variable 11.2 Integration of Scalar and Vector Functions - 11.3 Directional Derivatives and the Gradient Operator - 11.4 Conservative Fields and Potential Functions - 11.5 Divergence and Curl of a Vector - 11.6 Orthogonal Curvilinear Coordinates
• C H A P T E R 12 Vector Integral Calculus - 12.1 Background to Vector Integral Theorems - 12.2 Integral Theorems - 12.3 Transport Theorems - 12.4 Fluid Mechanics Applications of Transport Theorems - P A R T S I X COMPLEX ANALYSIS
• C H A P T E R 13 Analytic Functions - 13.1 Complex Functions and Mappings - 13.2 Limits, Derivatives, and Analytic Functions - 13.3 Harmonic Functions and Laplace’s Equation - and Branches 13.4 Elementary Functions, Inverse Functions,
  • C H A P T E R 14 Complex Integration
    • 14.1 Complex Integrals
      • Contour Integrals 14.2 Contours, the Cauchy–Goursat Theorem, and
    • 14.3 The Cauchy Integral Formulas
    • 14.4 Some Properties of Analytic Functions
      • Contour Integration C H A P T E R 15 Laurent Series, Residues, and
    • 15.1 Complex Power Series and Taylor Series
    • 15.2 Uniform Convergence
      • of Singularities 15.3 Laurent Series and the Classification
    • 15.4 Residues and the Residue Theorem
    • 15.5 Evaluation of Real Integrals by Means of Residues
  • C H A P T E R 16 The Laplace Inversion Integral
    • 16.1 The Inversion Integral for the Laplace Transform
      • to Boundary Value Problems C H A P T E R 17 Conformal Mapping and Applications
    • 17.1 Conformal Mapping
      • Value Problems 17.2 Conformal Mapping and Boundary
• P A R T S E V E N PARTIAL DIFFERENTIAL EQUATIONS
  • C H A P T E R 18 Partial Differential Equations
    • 18.1 What Is a Partial Differential Equation?
    • 18.2 The Method of Characteristics
    • 18.3 Wave Propagation and First Order PDEs
      • and Shocks 18.4 Generalizing Solutions: Conservation Laws
      • 18.5 The Three Fundamental Types of Linear Second Order PDE
        • Differential Equation for u ( x, y ) of a Second Order Constant Coefficient Partial
      • 18.7 Boundary Conditions and Initial Conditions
      • 18.8 Waves and the One-Dimensional Wave Equation
        • and Applications 18.9 The D’Alembert Solution of the Wave Equation
    • 18.10 Separation of Variables
    • 18.11 Some General Results for the Heat and Laplace Equation - Methods for PDEs 18.12 An Introduction to Laplace and Fourier Transform
• P A R T E I G H T NUMERICAL MATHEMATICS
  • C H A P T E R 19 Numerical Mathematics - 19.1 Decimal Places and Significant Figures - 19.2 Roots of Nonlinear Functions - 19.3 Interpolation and Extrapolation - 19.4 Numerical Integration - 19.5 Numerical Solution of Linear Systems of Equations - 19.6 Eigenvalues and Eigenvectors - 19.7 Numerical Solution of Differential Equations - Answers - References - Index

P R E F A C E

T

his book has evolved from lectures on engineering mathematics given regu- larly over many years to students at all levels in the United States, England, and elsewhere. It covers the more advanced aspects of engineering mathematics that are common to all first engineering degrees, and it differs from texts with similar names by the emphasis it places on certain topics, the systematic development of the underlying theory before making applications, and the inclusion of new material. Its special features are as follows.

Prerequisites

T

he opening chapter, which reviews mathematical prerequisites, serves two purposes. The first is to refresh ideas from previous courses and to provide basic self-contained reference material. The second is to remove from the main body of the text certain elementary material that by tradition is usually reviewed when first used in the text, thereby allowing the development of more advanced ideas to proceed without interruption.

Worked Examples

T

he numerous worked examples that follow the introduction of each new idea serve in the earlier chapters to illustrate applications that require relatively little background knowledge. The ability to formulate physical problems in mathemat- ical terms is an essential part of all mathematics applications. Although this is not a text on mathematical modeling, where more complicated physical applications are considered, the essential background is first developed to the point at which the physical nature of the problem becomes clear. Some examples, such as the ones involving the determination of the forces acting in the struts of a framed structure, the damping of vibrations caused by a generator and the vibrational modes of clamped membranes, illustrate important mathematical ideas in the context of practical applications. Other examples occur without specific applica- tions and their purpose is to reinforce new mathematical ideas and techniques as they arise. A different type of example is the one that seeks to determine the height of the tallest flagpole, where the height limitation is due to the phenomenon of

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buckling. Although the model used does not give an accurate answer, it provides a typical example of how a mathematical model is constructed. It also illustrates the reasoning used to select a physical solution from a scenario in which other purely mathematical solutions are possible. In addition, the example demonstrates how the choice of a unique physically meaningful solution from a set of mathematically possible ones can sometimes depend on physical considerations that did not enter into the formulation of the original problem.

Exercise Sets

T

he need for engineering students to have a sound understanding of mathe- matics is recognized by the systematic development of the underlying theory and the provision of many carefully selected fully worked examples, coupled with their reinforcement through the provision of large sets of exercises at the ends of sections. These sets, to which answers to odd-numbered exercises are listed at the end of the book, contain many routine exercises intended to provide practice when dealing with the various special cases that can arise, and also more chal- lenging exercises, each of which is starred, that extend the subject matter of the text in different ways. Although many of these exercises can be solved quickly by using standard computer algebra packages, the author believes the fundamental mathematical ideas involved are only properly understood once a significant number of exer- cises have first been solved by hand. Computer algebra can then be used with advantage to confirm the results, as is required in various exercise sets. Where computer algebra is either required or can be used to advantage, the exercise numbers are in blue. A comparison of computer-based solutions with those ob- tained by hand not only confirms the correctness of hand calculations, but also serves to illustrate how the method of solution often determines its form, and that transforming one form of solution to another is sometimes difficult. It is the author’s belief that only when fundamental ideas are fully understood is it safe to make routine use of computer algebra, or to use a numerical package to solve more complicated problems where the manipulation involved is pro- hibitive, or where a numerical result may be the only form of solution that is possible.

New Material

T

ypical of some of the new material to be found in the book is the matrix exponential and its application to the solution of linear systems of ordinary differential equations, and the use of the Green’s function. The introductory dis- cussion of the development of discontinuous solutions of first order quasilinear equations, which are essential in the study of supersonic gas flow and in vari- ous other physical applications, is also new and is not to be found elsewhere. The account of the Laplace transform contains more detail than usual. While the Laplace transform is applied to standard engineering problems, including

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Partial Differential Equations

A

n understanding of partial differential equations is essential in all branches of engineering, but accounts in engineering mathematics texts often fall short of what is required. This is because of their tendency to focus on the three standard types of linear second order partial differential equations, and their solution by means of separation of variables, to the virtual exclusion of first order equations and the systems from which these fundamental linear second order equations are derived. Often very little is said about the types of boundary and initial condi- tions that are appropriate for the different types of partial differential equations. Mention is seldom if ever made of the important part played by nonlinearity in first order equations and the way it influences the properties of their solutions. The account given here approaches these matters by starting with first order linear and quasilinear equations, where the way initial and boundary conditions and nonlinearity influence solutions is easily understood. The discussion of the effects of nonlinearity is introduced at a comparatively early stage in the study of partial differential equations because of its importance in subjects like fluid mechanics and chemical engineering. The account of nonlinearity also includes a brief discussion of shock wave solutions that are of fundamental importance in both supersonic gas flow and elsewhere. Linear and nonlinear wave propagation is examined in some detail because of its considerable practical importance; in addition, the way integral transform methods can be used to solve linear partial differential equations is described. From a rigorous mathematical point of view, the solution of a partial differential equation by the method of separation of variables only yields a formal solution, which only becomes a rigorous solution once the completeness of any set of eigenfunctions that arises has been established. To develop the subject in this manner would take the text far beyond the level for which it is intended and so the completeness of any set of eigenfunctions that occurs will always be as- sumed. This assumption can be fully justified when applying separation of vari- ables to the applications considered here and also in virtually all other practical cases.

Technology Projects

T

o encourage the use of technology and computer algebra and to broaden the range of problems that can be considered, technology-based projects have been added wherever appropriate; in addition, standard sets of exercises of a theoretical nature have been included at the ends of sections. These projects are not linked to a particular computer algebra package: Some projects illustrating standard results are intended to make use of simple computer skills while others provide insight into more advanced and physically important theoretical ques- tions. Typical of the projects designed to introduce new ideas are those at the end of the chapter on partial differential equations, which offer a brief introduc- tion to the special nonlinear wave solutions called solitons.

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Numerical Mathematics

A

lthough an understanding of basic numerical mathematics is essential for all engineering students, in a book such as this it is impossible to provide a sys- tematic account of this important discipline. The aim of this chapter is to provide a general idea of how to approach and deal with some of the most important and frequently encountered numerical operations, using only basic numerical techniques, and thereafter to encourage the use of standard numerical packages. The routines available in numerical packages are sophisticated, highly optimized and efficient, but the general ideas that are involved are easily understood once the material in the chapter has been assimilated. The accounts that are given here purposely avoid going into great detail as this can be found in the quoted references. However, the chapter does indicate when it is best to use certain types of routine and those circumstances where routines might be inappropriate. The details of references to literature contained in square brackets at the ends of sections are listed at the back of the book with suggestions for additional read- ing. An instructor’s Solutions Manual that gives outline solutions for the techno- logy projects is also available.

Acknowledgments

I

wish to express my sincere thanks to the reviewers and accuracy readers, those cited below and many who remain anonymous, whose critical comments and suggestions were so valuable, and also to my many students whose questions when studying the material in this book have contributed so fundamentally to its development. Particular thanks go to:

Chun Liu, Pennsylvania State University William F. Moss, Clemson University Donald Hartig, California Polytechnic State University at San Luis Obispo Howard A. Stone, Harvard University Donald Estep, Georgia Institute of Technology Preetham B. Kumar, California State University at Sacramento Anthony L. Peressini, University of Illinois at Urbana-Champaign Eutiquio C. Young, Florida State University Colin H. Marks, University of Maryland Ronald Jodoin, Rochester Institute of Technology Edgar Pechlaner, Simon Fraser University Ronald B. Guenther, Oregon State University Mattias Kawski, Arizona State University L. F. Shampine, Southern Methodist University

In conclusion, I also wish to thank my editor, Barbara Holland, for her invalu- able help and advice on presentation; Julie Bolduc, senior production editor, for her patience and guidance; Mike Sugarman, for his comments during the early stages of writing; and, finally, Chuck Glaser, for encouraging me to write the book in the first place.

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