modern engineering mathematics
DESCRIPTION
ContentTRANSCRIPT
ModernEngineeringMathematics
Fourth Edition
Glyn Jamesand ; 'David BurleyDick ClementsPhil DykeJohn SearlJerry Wright
Coventry University
University of SheffieldUniversity of BristolUniversity of PlymouthUniversity of EdinburghAT&T Shannon Laboratory
PEARSON
PrenticeHall
Harlow, England • London • New York • Boston • San Francisco • Toronto • Sydney • Singapore • Hong KongTokyo • Seoul • Taipei • New Delhi • Cape Town • Madrid • Mexico City • Amsterdam • Munich • Paris • Milan
tr. •
Contents
Preface :About the authors
XXI
xxiv
Chapter 1 Numbers, Algebra and Geometry
1.1 Introduction
1*2 Number and arithmetic1.2.1 Number line1.2.2 Rules of arithmetic1.2.3 Exercises (1-5)1.2.4* Inequalities1.2.5 Modulus and intervals1.2.6 Exercises (6-10)
223778
11
1.3 Algebra '1.3.1 Algebraic manipulation1.3.2 Exercises (11-16)1.3.3 Equations, inequalities and identities1.3.4 Exercises (17-28) ; '1.3.5 Suffix, sigma and pi notation1.3.6 Factorial notation and the binomial expansion1.3.7 Exercises (29-31)
1.4 Geometry1.4.1 Coordinates1.4.2 Straight lines ;1.4.3 Circles
•,. 1.4.4 Exercises (32-38)1.4.5 Conies1.4.6 Exercises (39-41)
1213192027273033
Numbers and accuracyt5.1 Representation of numbers1.5,2 Rounding, decimal places and significant figures
33.333335383844
444547
Vi CONTENTS
1.5.3 Estimating the effect of rounding errors1.5.4 Exercises (42-55)1.5.5 Computer arithmetic1.5.6 Exercises (56-58)
49545556
1.6 Engineering applications 56
1.7 Review exercises (1-25) 59
Chapter 2 Functions
2.1 Introduction 64
2.2 Basic definitions : :
2.2.1 Concept of a function2.2.2 Exercises (1-6)2.2.3 Inverse functions2.2.4 Composite functions .2.2.5 Exercises (7-13)2.2.6 Odd, even and periodic functions2.2.7 Exercises (14-15)
64
64737478818287
2.3 Linear and quadratic functions 87
2.3.1 Linear functions 872.3.2 Least squares fit of a linear function to experimental data 892.3.3 Exercises (16-22) 932.3.4 The quadratic function 942.3.5 Exercises (23-28) 97
2.4 Polynomial functions
2.4.1 Basic properties2.4.2 Factorization2.4.3 Nested multiplication and synthetic division2.4.4 Roots of polynomial equations2.4.5 Exercises (29-37)
2.5 Rational functions
2.5.1 Partial fractions2.5.2 Exercises (38-41)2.5.3 Asymptotes2.5.4 Parametric representation2.5.5 Exercises (42-46)
9899100102105112
114
116122123126128
2.6 Circular functions
2.6.1 Trigonometric ratios2.6.2 Exercises (47-53)
128129131
CONTENTS Vii _--s"
2.6.3 Circular functions2.6.4 Trigonometric identities2.6.5 Amplitude and phase '2.6.6 Exercises (54-65)2.6.7 Inverse circular (trigonometric) functions2.6.8 Polar coordinates2.6.9 Exercises (66-70)
132138142145146148151
2.7 Exponential, logarithmic and hyperbolic functions
2.7.1 Exponential functions2.7.2 Logarithmic functions,2.7.3 Exercises (71-79) ; '.2.7A Hyperbolic functions2.7.5 Inverse hyperbolic functions2.7.6 Exercises (80-87)
152
152155157157162164
2.8 Irrational functions
2.8.1 Algebraic functions2.8.2 Implicit functions2.8.3 Piecewise defined functions2.8.4 Exercises (88-97)
164
165166170172
2.9 Numerical evaluation of functions
2.9.1 Tabulated functions and interpolation2.9.2 Exercises (98-103) :
173
174178
2.10 Engineering application: a design problem 179
2.11 Review exercises (1-23) 181
Chapter 3 Complex Numbers
3.1 Introduction 185
3.2 Properties
3.2.1 The Argand diagram3.2.2 The arithmetic of complex numbers3.2.3 Complex conjugate3.2.4 Modulus and argument3.2.5 Exercises (1-14)3.2.6 Polar form of a complex number3.2.7 Euler's formula
186
186187190191195196200
VIII CONTENTS
3.2.8 Exercises (15-18) 2023.2.9 Relationship between circular and hyperbolic functions 2023.2.10 Logarithm of a complex number 2063.2.11 Exercises (19-24) 207
3.3 Powers of complex numbers 208
3.3.1 De Moivre's theorem 2083.3.2 Powers of trigonometric functions and multiple angles 2123.3.3 Exercises (25-32) 215
3.4 Loci in the complex plane 216
3.4.1 Straight lines • 2163.4.2 Circles • 2173.4.3 More general loci 2193.4.4 Exercises (33-41) 220_
3.5 Functions of a complex variable 221
3.5.1 Exercises (42-45) , 223
3.6 Engineering application: alternating currents in electrical networks 223
.3.6.1 Exercises (46-47) : _225
3.7 Review'exercises (1-34) : 225
Chapter 4 Vector Algebra
4.1 Introduction
4.2 Basic definitions and results
4.2.1 Cartesian coordinates4.2.2 Scalars and vectors4.2.3 Addition of vectors4.2.4 Cartesian components and basic properties4.2.5 Complex numbers as vectors4.2.6 Exercises (1-16)4.2.7, The scalar product4.2.8 Exercises (17-30)4.2.9 The vector product4.2.10 Exercises (31-42)4.2.11 Triple products4.2.12 Exercises (43-51)
230
231
231233235241247249251257258268269275
CONTENTS ix
Iff 4.3 The vector treatment of the geometry of lines and planes
4.3.1 Vector equation of a line4.3.2 Vector equation of a plane4.3.3 Exercises (52-67)
4.4 Engineering application: spin-dryer suspension
4.4.1 Point-particle model
4.5 Engineering application: cable stayed bridge
4.5.1 A simple stayed bridge
276
276283286
287
287
290
290
4.6 Review exercises (1-24) 292
Chapter 5 Matrix Algebra
5.1 Introduction 297
5,2 Definitions and properties
5.2.1 Definitions5.2.2 Basic operations of matrices5.2.3 Exercises (1-10) •5.2.4 Matrix multiplication5.2.5 Exercises (11-16)5.2.6 Properties of matrix multiplication5.2.7 Exercises (17-33)
5.3 Determinants
5.3.1 Exercises (34-50)
5.4 The inverse matrix
5.4.1 Exercises (51-59)
5.5 Linear equations
5.5.1 Exercises (60-71)5.5.2 The solution of linear equations: elimination methods5.5.3 Exercises (72-80)5.5.4 The solution of linear equations: iterative methods5.5.5 Exercises (81-86)
5.6 Rank
5.6.1 Exercises (87-95)
299
301304308310314315325
328
340
341
345
347
354356369371377
377
385
X CONTENTS
5.7 The eigenvalue problem
5.7.1 The characteristic equation5.7.2 Eigenvalues and eigenvectors5.7.3 Exercises (96-97)5.7.4 Repeated eigenvalues5.7.5 Exercises (98-102)5.7.6 Some useful properties of eigenvalues5.7.7 Symmetric matrices5.7.8 Exercises (103-107)
387
387389395396400400402403
5.8 Engineering application: spring systems
5.8.1 A two-particle system5.8.2 An n-particle system
5.9
403
404404
Engineering application: steady heat transfer throughcomposite materials
5.9.1 Introduction5.9.2 Heat conduction5.9.3 The three-layer situation5.9.4 Many-layer situation
407
407408408410
5.10 Review exercises (1-26) 411
Chapter 6 An Introduction to Discrete Mathematics
6.1 Introduction 418
6.2 Set theory
6.2.1 Definitions and notation6.2.2 Union and intersection6.2.3 Exercises (1-8)6.2.4 Algebra of sets6.2.5 Exercises (9-17)
418419420422422427
6.3 Switching and logic circuits
6.3.1 Switching circuits6.3.2 Algebra of switching circuits6.3.3 Exercises (18-29)6.3.4 Logic circuits6.3.5 Exercises (30-31)
429429430436437441
6.4 Propositional logic and methods of proof
6.4.1 Propositions6.4.2 Compound propositions6.4.3 Algebra of statements
442
442444447
CONTENTS XI
6.4.4 Exercises (32-37)6.4.5 Implications and proofs6.4.6 Exercises (38-4:7)
450450456
6.5 Engineering application: expert systems
6.6 Engineering application: control
6.7 Review exercises (1-23)
457
459
462
Chapter 7 Sequences, Series and Limits
7.1 Introduction 467
7.2 Sequences and series
7.2.1 Notation7.2.2 Graphical representation of sequences7.2.3 Exercises (1-13)
467
467469472
7.3 Finite sequences and series ,:.
7.3.1 Arithmetical sequences and series7.3.2^ Geometric sequences and series7.3.3 Other finite series7.3.4 Exercises (14-25)
474
474475477480
7.4 Recurrence relations
7.4.1 First-order linear recurrence relations withconstant coefficients
7.4.2 Exercises (26-28) .7.4.3 Second-order linear recurrence relations with
constant coefficients7.4.4 Exercises (29-35)
481
482485
486494
7.5 Limit of a sequence
7.5.1 Convergent sequences7.5.2 Properties of convergent sequences7.5.3 Computation of limits7.5.4 Exercises (36-40)
494
495497499501
7.6 Infinite series
7.6.1 Convergence of infinite series7.6.2 Testa for convergence of positive series7.6.3 The absolute convergence of general series7.6.4 Exercises (41-49)
502
502504507508
Xii CONTENTS
7.7 Power series
7.7.1 Convergence of power series7.7.2 Special power series7.7.3 Exercises (50-56)
7.8 Functions of a real variable
7.8.1 Limit of a function of a real variable7.8.2 One-sided limits7.8.3 Exercises (57-61)
7.9 Continuity of functions of a real variable
7.9.1 Properties of continuous functions7.9.2 Continuous and discontinuous functions7.9.3 Numerical location of zeros7.9.4 Exercises (62-69)
7.10 Engineering application: insulator chain
509
509511517
518
518522524
525
525527529532
532
7.11 Engineering application: approximating functions andPade approximants 533
7.12 Review exercises (1-25) 535
Chapter 8 Differentiation and Integration
8.1 Introduction
8.2 Differentiation
8.2.1 Rates of change8.2.2 Definition of a derivative8.2.3 Interpretation as the slope of a tangent8.2.4 Differentiate functions8.2.5 Speed, velocity and acceleration8.2.6 Exercises (1-7)8.2.7 Mathematical modelling using derivatives8.2.8 Exercises (8-18)
8.3 Techniques of differentiation
8.3.1 Basic rules of differentiation8.3.2 Derivative of xr
8.3.3 Exercises (19-23)8.3.4 Differentiation of polynomial functions
540
541
541542544546547548549556
557
558560564564
CONTENTS xiii
8.3.5 Differention of rational functions8.3.6 Differentiation of composite functions8.3.7 Differentiation of inverse functions8.3.8 Exercises (24-31)8.3.9 Differentiation of circular functions8.3.10 Extended form of the chain rule8.3.11 Exercises (32-34)8.3.12 Differentiation of exponential and related functions8.3.13 Exercises (35-43)8.3.14 Parametric and implicit differentiation8.3.15 Exercises (44-54)
8.4 Higher derivatives
8.4.1 The second derivative8.4.2 Exercises (55-67)8.4.3 Curvature of plane curves8.4.4 Exercises (68-71)
567568573574575579581581586586591
592
592596597600
8.5 Applications to optimization problems
8.5.1 Optimal values8.5.2 Exercises (72-81)
600
600609
8.6 Numerical differentiation
8.6.1 The chord approximation8.6.2 Exercises (82-86)
611
611613
8.7 Integration
8.7.1 Basic ideas and definitions8.7.2 Mathematical modelling using integration8.7.3 Exercises (87-95)8.7.4 Definite and indefinite integrals8.7.5 The Fundamental Theorem of Calculus8.7.6 Exercise (96)
8.8 Techniques of integration
8.8.1 Integration as antiderivative8.8.2 Exercises (97-104)8.8.3 Integration by parts8.8.4 Exercises (105-107)8.8.5 Integration by substitution8.8.6 Exercises (108-116)
8.9 Applications of integration
8.9.1 Volume of a solid of revolution8.9.2 Centroid of a Diane area
613
613616620620623625
625
625636637640640645
646
646647
Xiv CONTENTS
8.9.3 Centre of gravity of a solid of revolution 6498.9.4 Mean values . 6498.9.5 Root mean square values 6508.9.6 Arclength and surface area 6508.9.7 Exercises (117-125) 656
8.10 Numerical evaluation of integrals 657
8.10.1 The trapezium rule 6578.10.2 Simpson's rule 6638.10.3 Exercises (126-131) \ • 666
8.11 Engineering application: design of prismatic channels 667
8.12 Engineering application: harmonic analysis of periodic functions 669
8.13 Review exercises (1-39) 671
Chapter 9 Further Calculus
9.1 Introduction
9.2 Improper integrals
9.2.1 Integrand with an infinite discontinuity9.2.2 Infinite integrals9.2.3 Exercise (1)
9.3 Some theorems with applications to numerical methods
9.3.1 Rolle's theorem and the first mean value theorems9.3.2 Convergence of iterative schemes9,3.3 Exercises (2-7)
9.4 Taylor's theorem and related results
9.4.1 Taylor polynomials and Taylor's theorem9.4.2 Taylor and Maclaurin series9.4.3 L'Hopital's rule9.4.4 Exercises (8-20)9.4.5 Interpolation revisited9.4.6 Exercises (21-23)9A7 The convergence of iterations revisited9.4.8 Newton-Raphson procedure9.4.9 Optimization revisited9.4.10 Exercises (24-27)9.4.11 Numerical integration9.4.12 Exercises (28-31)
680
680
681684685
686
686689693
693
693696701702
: 703704705706709709709711
mBHI-
m.BfflBSi
iHm
WI
9.5 Calculus of vectors
9.5.1 Differentiation and integration of vectors9.5.2 Exercises (32-36)
9.6 Functions of several variables
9.6.1 Representation of functions of two variables9.6.2 Partial derivatives9.6.3 Directional derivatives9.6.4 Exercises (37-46)9.6.5 The chain rule ;9.6.6 Exercises (47-55)9.6.7 Successive differentiation9.6.8 Exercises (56-64)9.6.9 The total differential and small errors9.6.10 Exercises (65-72)9.6.11 Exact differentials9.6.12 Exercises (73-75)
9.7 Taylor's theorem for functions of two variables
9.7.1 Taylor's theorem9.7.2 Optimization of unconstrained functions9.7.3 Exercises (76-84)9.7.4 Optimization of constrained functions9.7.5 Exercises (85-90)
9.8 Engineering application: deflection of a built-in column
9.9 Engineering application: streamlines in fluid dynamics
9.10 Review exercises (1-35)
CONTENTS XV
712
712714
715
715717721724725729729733733736737739
739
740743748749753
754
756
759
Chapter 10 Introduction to Ordinary Differential Equations
10.1 Introduction
10.2 Engineering examples
10.2.1 The take-off run of an aircraft10.2.2 Domestic hot-water supply10.2.3 Hydro-electric power generation10.2.4 Simple electrical circuits
765
765
765767768769
10.3 The classification of differential equations
10.3.1 Ordinary and partial differential equations10.3.2 Independent and dependent variables
770
771771
XVI CONTENTS
10.4
10.5
10.6
10.7
10.8
10.9
10.3.3 The order of a differential equation10.3.4 Linear and nonlinear differential equations10.3.5 Homogeneous and nonhomogeneous equations10.3.6 Exercises (1-2)
Solving differential equations
10.4.1 Solution by inspection10.4.2 General and particular solutions10.4.3 Boundary and initial conditions10.4.4 Analytical and numerical solution10.4.5 Exercises (3-6)
First-order ordinary differential equations
10.5.1 A geometrical perspective10.5.2 Exercises (7-10)10.5.3 Solution of separable differential equations10.5.4 Exercises (11-17)
dx f x ~\10.5.5 Solution of differential equations of —- = fl - form
10.5.6 Exercises (18-22)10.5.7 Solution of exact differential equations10.5.8 Exercises (23-30)10.5.9 Solution of linear differential equations10.5.10 Solution of the Bernoulli differential equations10.5.11 Exercises (31-38)
Numerical solution of first-order ordinary differential equations
10.6.1 A simple solution method: Euler's method10.6.2 Analysing Euler's method10.6.3 Using numerical methods to solve engineering problems10.6.4 Exercises (39-45)
Engineering application: analysis of damper performance
Linear differential equations
10.8.1 Differential operators10.8.2 Linear differential equations10.8.3 Exercises (46-54)
Linear constant-coefficient differential equations
10.9.1 Linear homogeneous constant-coefficient equations10.9.2 Exercises (55-61)10.9.3 Linear nonhomogeneous constant-coefficient equations10.9.4 Exercises (62-65)
772773774775
776
776111118781782
783
783786786788
789
791791794795799801
802
803805808810
811
816
816818824
826
826831832838
CONTENTS XVII
10.10 Engineering application: second-order linear constant-coefficientdifferential equations • , 839
10.10.1 Free oscillations of elastic systems 83910.10.2 Free oscillations of damped elastic systems 84310.10.3 Forced oscillations of elastic systems 84610.10.4 Oscillations in electrical circuits ' 85010.10.5 Exercises (66-73) 851
10.11 Numerical solution of seconds arid higher-orderdifferential equations 853
10.11.1 Numerical solution of coupled first-order equations 85310.11.2 State-space representation of higher-order systems 85610.11.3 Exercises (74-79) : 859
10.12 Qualitative analysis of second-order differential equations
10.12.1 Phase-plane plots10.12.2 Exercises (80-81)
861861865
10.13 Review exercises (1-35) 866
Chapter 11 Introduction to Laplace Transforms
11.1 Introduction 874
1.1.2 The Laplace transform 876
11.2.1 Definition and notation 87611.2.2 Transforms of simple functions 87811.2.3 Existence of the Laplace transform ; 88111.2.4 Properties of the Laplace transform 88311.2.5 Table of Laplace transforms 89111.2.6 Exercises (1-3) 89211.2.7 The inverse transform 89211.2.8 Evaluation of inverse transforms 89311.2.9 Inversion using the first shift theorem 89511.2.10 Exercise (4) 897
11.3 Solution of differential equations 897
11.3.1 Transforms of derivatives 89711.3.2 Transforms of integrals 89911.3.3 Ordinary differential equations 90011.3.4 Exercise (5) 90611.3.5 Simultaneous differential equations 90711.3.6 Exercise (6) • 909
xviii CONTENTS
11.4 Engineering applications; electrical circuits and; mechanical vibrations
11.4.1 Electrical circuits11.4.2 Mechanical vibrations :11.4.3 Exercises (7-12) ;
910910915919
11.5 Review exercises (1-18) 920
Chapter 12 Introduction to Fourier Series
12.1 Introduction 925
12.2 Fourier series expansion
12.2.1 Periodic functions : :
12.2.2 Fourier's theorem ,12.2.3 The Fourier coefficients12.2.4 Functions of period:2rr • ' ,'12.2.5 Even and odd functions . ';.12.2.6 Even and odd harmonics ;
12.2.7 Linearity property12.2.8 Convergence of the Fourier series12.2.9 Exercises (1-7)12.2.10 Functions of period T12.2.11 Exercises (8-13)
926926927928931938942944946949951953
12.3 Functions defined over a finite interval
12.3.1 Full-range series12.3.2 Half-range cosine and sine series12.3.3 Exercises (14-23)
954954956960
12.4 Differentiation and integration of Fourier series
12.4.1 Integration of:a Fourier series12.4.2 Differentiation of a Fourier series12.4.3 Exercises (24-26)
961961964965
12.5 Engineering application: analysis of aslider-crank mechanism 966
12.6 Review exercises (1-21.) 969
CONTENTS Xix
Chapter 13 Data Handling and Probability Theory
13.1 Introduction" 974
13.2 The raw material of statistics
13.2.1 Experiments and sampling13.2.2 Histograms of data13.2.3 Alternative types of plot13.2.4 Exercises (1-5)
975975975978980
13.3 Probabilities of random events
13.3.1 Interpretations, of probability13.3.2 Sample space and events13.3.3 Axioms, of probability13.3.4 Conditional probability13.3.5 Independence13.3.6 Exercises (6-23)
980
980981982984988991
13.4 Random variables
13.4.1 Introduction and definition13.4.2 Discrete random variables13.4.3 Continuous random variables13.4.4 • Properties of density and distribution functions13.4.5 Exercises (24-31)13.4.6 •• Measures of location and dispersion13.4.7 Expected values13.4.8 Independence of random variables13.4.9 Scaling and adding random variables13.4.10 Measures from sample data13.4.11 Exercises (32-48)
992
992993994995998998
10021003100410071011
13.5 Important practical distributions
13.5.1 The binomial distribution13.5.2 The Poisson distribution •.13.5.3 The normal distribution13.5.4 The central limit theorem13.5.5 Normal approximation to the binomial13.5.6 Random variables for simulation13.5.7 Exercises (49-65) ••'
1013
1013101510181021102410261027
13.6 Engineering application: quality control
13.6-1 Attribute control charts13.6.2 United States standard attribute charts13.6.3 Exercises (66-67)
1029
102910311032
XX CONTENTS
13.7 Engineering application: clustering of rare events
13.7.1 Introduction13.7.2 Survey of near-misses between aircraft13.7.3 Exercises (68-69)
1032103210331035
13.8 Review exercises (1-1.3) 1035
Appendix I Tables
Al. l Some useful resultsAI.2 Trigonometric identitiesAI.3 Derivatives and integralsAI.4 Some useful standard integrals
1038
1038104110421043
Answers to Exercises 1044
Index 1082