Jeffrey - Advanced Engineering Mathematics (Aca..

1155 Pages·2007·8.1 MB·English

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
1
CHAPTER 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
vii
P A R T T W O VECTORS AND MATRICES 53
2
CHAPTER Vectors and Vector Spaces 55
2.1 Vectors, Geometry, and Algebra 56
2.2 The Dot Product (Scalar Product) 70
2.3 The Cross Product (Vector Product) 77
2.4 Linear Dependence and Independence of Vectors
and Triple Products 82
2.5 n-Vectors and the Vector Space Rn 88
2.6 Linear Independence, Basis, and Dimension 95
2.7 Gram–Schmidt Orthogonalization Process 101
3
CHAPTER Matrices and Systems of Linear Equations 105
3.1 Matrices 106
3.2 Some Problems That Give Rise to Matrices 120
3.3 Determinants 133
3.4 Elementary Row Operations, Elementary Matrices,
and Their Connection with Matrix Multiplication 143
3.5 The Echelon and Row-Reduced Echelon
Forms of a Matrix 147
3.6 Row and Column Spaces and Rank 152
3.7 The Solution of Homogeneous Systems
of Linear Equations 155
3.8 The Solution of Nonhomogeneous Systems
of Linear Equations 158
3.9 The Inverse Matrix 163
3.10 Derivative of a Matrix 171
4
CHAPTER Eigenvalues, Eigenvectors, and Diagonalization 177
4.1 Characteristic Polynomial, Eigenvalues,
and Eigenvectors 178
4.2 Diagonalization of Matrices 196
4.3 Special Matrices with Complex Elements 205
4.4 Quadratic Forms 210
4.5 The Matrix Exponential 215
viii
P A R T T H R E E ORDINARY DIFFERENTIAL EQUATIONS 225
5
CHAPTER 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
6
CHAPTER 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(cid:4)(cid:4) + 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
ix
7
CHAPTER The Laplace Transform 379
7.1 Laplace Transform: Fundamental Ideas 379
7.2 Operational Properties of the Laplace Transform 390
7.3 Systems of Equations and Applications of the
Laplace Transform 415
7.4 The Transfer Function, Control Systems, and Time Lags 437
8
CHAPTER SeriesSolutionsofDifferentialEquations,Special
Functions, and Sturm–Liouville Equations 443
8.1 A First Approach to Power Series Solutions
of Differential Equations 443
8.2 A General Approach to Power Series Solutions
of Homogeneous Equations 447
8.3 Singular Points of Linear Differential Equations 461
8.4 The Frobenius Method 463
8.5 The Gamma Function Revisited 480
8.6 Bessel Function of the First Kind J (x) 485
n
8.7 Bessel Functions of the Second Kind Yν(x) 495
8.8 Modified Bessel Functions Iν(x) and Kν(x) 501
8.9 A Critical Bending Problem: Is There a Tallest Flagpole? 504
8.10 Sturm–Liouville Problems, Eigenfunctions,
and Orthogonality 509
8.11 Eigenfunction Expansions and Completeness 526
P A R T F O U R FOURIER SERIES, INTEGRALS, AND
THE FOURIER TRANSFORM 543
9
CHAPTER Fourier Series 545
9.1 Introduction to Fourier Series 545
9.2 Convergence of Fourier Series and Their Integration
and Differentiation 559
9.3 Fourier Sine and Cosine Series on 0≤x ≤L 568
9.4 Other Forms of Fourier Series 572
9.5 Frequency and Amplitude Spectra of a Function 577
9.6 Double Fourier Series 581
x
10
CHAPTER Fourier Integrals and the Fourier Transform 589
10.1 The Fourier Integral 589
10.2 The Fourier Transform 595
10.3 Fourier Cosine and Sine Transforms 611
P A R T F I V E VECTOR CALCULUS 623
11
CHAPTER Vector Differential Calculus 625
11.1 Scalar and Vector Fields, Limits, Continuity,
and Differentiability 626
11.2 Integration of Scalar and Vector Functions
of a Single Real Variable 636
11.3 Directional Derivatives and the Gradient Operator 644
11.4 Conservative Fields and Potential Functions 650
11.5 Divergence and Curl of a Vector 659
11.6 Orthogonal Curvilinear Coordinates 665
12
CHAPTER Vector Integral Calculus 677
12.1 Background to Vector Integral Theorems 678
12.2 Integral Theorems 680
12.3 Transport Theorems 697
12.4 Fluid Mechanics Applications of Transport Theorems 704
P A R T S I X COMPLEX ANALYSIS 709
13
CHAPTER Analytic Functions 711
13.1 Complex Functions and Mappings 711
13.2 Limits, Derivatives, and Analytic Functions 717
13.3 Harmonic Functions and Laplace’s Equation 730
13.4 Elementary Functions, Inverse Functions,
and Branches 735
xi
14
CHAPTER Complex Integration 745
14.1 Complex Integrals 745
14.2 Contours, the Cauchy–Goursat Theorem, and
Contour Integrals 755
14.3 The Cauchy Integral Formulas 769
14.4 Some Properties of Analytic Functions 775
15
CHAPTER Laurent Series, Residues, and
Contour Integration 791
15.1 Complex Power Series and Taylor Series 791
15.2 Uniform Convergence 811
15.3 Laurent Series and the Classification
of Singularities 816
15.4 Residues and the Residue Theorem 830
15.5 Evaluation of Real Integrals by Means of Residues 839
16
CHAPTER The Laplace Inversion Integral 863
16.1 The Inversion Integral for the Laplace Transform 863
17
CHAPTER Conformal Mapping and Applications
to Boundary Value Problems 877
17.1 Conformal Mapping 877
17.2 Conformal Mapping and Boundary
Value Problems 904
P A R T S E V E N PARTIAL DIFFERENTIAL EQUATIONS 925
18
CHAPTER Partial Differential Equations 927
18.1 What Is a Partial Differential Equation? 927
18.2 The Method of Characteristics 934
18.3 Wave Propagation and First Order PDEs 942
18.4 Generalizing Solutions: Conservation Laws
and Shocks 951
xii
18.5 The Three Fundamental Types of Linear Second Order PDE 956
18.6 Classification and Reduction to Standard Form
of a Second Order Constant Coefficient Partial
Differential Equation for u(x,y) 964
18.7 Boundary Conditions and Initial Conditions 975
18.8 Waves and the One-Dimensional Wave Equation 978
18.9 The D’Alembert Solution of the Wave Equation
and Applications 981
18.10 Separation of Variables 988
18.11 Some General Results for the Heat and Laplace Equation 1025
18.12 An Introduction to Laplace and Fourier Transform
Methods for PDEs 1030
P A R T E I G H T NUMERICAL MATHEMATICS 1043
19
CHAPTER Numerical Mathematics 1045
19.1 Decimal Places and Significant Figures 1046
19.2 Roots of Nonlinear Functions 1047
19.3 Interpolation and Extrapolation 1058
19.4 Numerical Integration 1065
19.5 Numerical Solution of Linear Systems of Equations 1077
19.6 Eigenvalues and Eigenvectors 1090
19.7 Numerical Solution of Differential Equations 1095
Answers 1109
References 1143
Index 1147
xiii
P R E F A C E
This book has evolved from lectures on engineering mathematics given regu-
larlyovermanyyearstostudentsatalllevelsintheUnitedStates,England,and
elsewhere.Itcoversthemoreadvancedaspectsofengineeringmathematicsthat
arecommontoallfirstengineeringdegrees,anditdiffersfromtextswithsimilar
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.Itsspecialfeaturesareasfollows.
Prerequisites
The 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
bodyofthetextcertainelementarymaterialthatbytraditionisusuallyreviewed
whenfirstusedinthetext,therebyallowingthedevelopmentofmoreadvanced
ideastoproceedwithoutinterruption.
Worked Examples
Thenumerousworkedexamplesthatfollowtheintroductionofeachnewidea
serveintheearlierchapterstoillustrateapplicationsthatrequirerelativelylittle
backgroundknowledge.Theabilitytoformulatephysicalproblemsinmathemat-
icaltermsisanessentialpartofallmathematicsapplications.Althoughthisisnot
atextonmathematicalmodeling,wheremorecomplicatedphysicalapplications
areconsidered,theessentialbackgroundisfirstdevelopedtothepointatwhich
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
contextofpracticalapplications.Otherexamplesoccurwithoutspecificapplica-
tionsandtheirpurposeistoreinforcenewmathematicalideasandtechniquesas
theyarise.
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
xv