Advanced Modern Engineering Mathematics, 4th Edition

Cover Page......Page 1
Title Page......Page 4
ISBN 9780273719236......Page 5
Advanced Modern Engineering Mathematics......Page 2
Contents......Page 6
Preface......Page 20
About the Authors......Page 22
Publisher’s Acknowledgements......Page 24
Matrix Analysis......Page 26
Review of matrix algebra......Page 27
Basic operations on matrices......Page 28
Adjoint and inverse matrices......Page 30
Linear equations......Page 32
Rank of a matrix......Page 34
Vector spaces......Page 35
Linear independence......Page 36
Transformations between bases......Page 37
The eigenvalue problem......Page 39
The characteristic equation......Page 40
Eigenvalues and eigenvectors......Page 42
Repeated eigenvalues......Page 48
Some useful properties of eigenvalues......Page 52
Symmetric matrices......Page 54
The power method......Page 55
Gerschgorin circles......Page 61
Exercises (14-19)......Page 63
Reduction to diagonal form......Page 64
The Jordan canonical form......Page 67
Exercises (20-27)......Page 71
Quadratic forms......Page 72
Exercises (28-34)......Page 78
Functions of a matrix......Page 79
Exercises (35-42)......Page 90
Singular value decomposition......Page 91
Singular values......Page 93
Singular value decomposition (SVD)......Page 97
Pseudo inverse......Page 100
Exercises (43-50)......Page 106
Single-input–single-output (SISO) systems......Page 107
Multi-input–multi-output (MIMO) systems......Page 112
Exercises (51-55)......Page 113
Direct form of the solution......Page 114
The transition matrix......Page 116
Evaluating the transition matrix......Page 117
Exercises (56-61)......Page 119
Spectral representation of response......Page 120
Canonical representation......Page 123
Exercises (62-68)......Page 128
Engineering application: Lyapunov stability analysis......Page 129
Exercises (69-73)......Page 131
Engineering application: capacitor microphone......Page 132
Review exercises (1–20)......Page 136
Numerical Solution of Ordinary Differential Equations......Page 140
Engineering application: motion in a viscous fluid......Page 141
Numerical solution of first-order ordinary differential equations......Page 142
A simple solution method: Euler’s method......Page 143
Analysing Euler’s method......Page 147
Using numerical methods to solve engineering problems......Page 150
Exercises (1-7)......Page 152
More accurate solution methods: multistep methods......Page 153
Local and global truncation errors......Page 159
More accurate solution methods: predictor–corrector methods......Page 161
More accurate solution methods: Runge–Kutta methods......Page 166
Exercises (8-17)......Page 170
Stiff equations......Page 172
Computer software libraries and the `state of the art'......Page 174
Numerical solution of coupled first-order equations......Page 176
State-space representation of higher-order systems......Page 181
Exercises (18-23)......Page 185
Boundary-value problems......Page 186
The method of shooting......Page 187
Function approximation methods......Page 189
Engineering application: oscillations of a pendulum......Page 195
Engineering application: heating of an electrical fuse......Page 199
Review exercises (1–12)......Page 204
Vector Calculus......Page 206
Introduction......Page 207
Basic concepts......Page 208
Exercises (1-10)......Page 216
Transformations......Page 217
Exercises (11-17)......Page 220
The total differential......Page 221
The gradient of a scalar point function......Page 224
Derivatives of a vector point function......Page 228
Divergence of a vector field......Page 229
Curl of a vector field......Page 231
Further properties of the vector operator ∇......Page 235
Topics in integration......Page 239
Line integrals......Page 240
Exercises (56-64)......Page 243
Double integrals......Page 244
Exercises (65-76)......Page 249
Green’s theorem in a plane......Page 250
Exercises (77-82)......Page 254
Surface integrals......Page 255
Volume integrals......Page 262
Exercises (92-102)......Page 265
Gauss’s divergence theorem......Page 266
Stokes’ theorem......Page 269
Exercises (103-112)......Page 272
Engineering application: streamlines in fluid dynamics......Page 273
Engineering application: heat transfer......Page 275
Review exercises (1–21)......Page 279
Functions of a Complex Variable......Page 282
Introduction......Page 283
Complex functions and mappings......Page 284
Linear mappings......Page 286
Inversion......Page 293
Bilinear mappings......Page 298
Exercises (9-19)......Page 304
The mapping w = z2......Page 305
Complex differentiation......Page 307
Cauchy–Riemann equations......Page 308
Conjugate and harmonic functions......Page 313
Mappings revisited......Page 315
Exercises (33-37)......Page 319
Power series......Page 320
Taylor series......Page 324
Exercises (40-43)......Page 327
Laurent series......Page 328
Singularities and zeros......Page 333
Residues......Page 336
Exercises (50-52)......Page 341
Contour integrals......Page 342
Cauchy’s theorem......Page 345
Exercises (53-59)......Page 352
The residue theorem......Page 353
Evaluation of definite real integrals......Page 356
Exercises (60-65)......Page 359
Engineering application: analysing AC circuits......Page 360
A heat transfer problem......Page 361
Current in a field-effect transistor......Page 363
Exercises (66-72)......Page 366
Review exercises (1–24)......Page 367
Laplace Transforms......Page 370
Introduction......Page 371
Definition and notation......Page 373
Transforms of simple functions......Page 375
Existence of the Laplace transform......Page 378
Properties of the Laplace transform......Page 380
Table of Laplace transforms......Page 388
The inverse transform......Page 389
Evaluation of inverse transforms......Page 390
Inversion using the first shift theorem......Page 392
Exercise (4)......Page 394
Transforms of derivatives......Page 395
Transforms of integrals......Page 396
Ordinary differential equations......Page 397
Simultaneous differential equations......Page 403
Exercises (5-6)......Page 405
Engineering applications: electrical circuits and mechanical vibrations......Page 406
Electrical circuits......Page 407
Mechanical vibrations......Page 411
Exercises (7-12)......Page 415
The Heaviside step function......Page 417
Laplace transform of unit step function......Page 420
The second shift theorem......Page 422
Inversion using the second shift theorem......Page 425
Differential equations......Page 428
Periodic functions......Page 432
Exercises (13-24)......Page 436
The impulse function......Page 438
The sifting property......Page 439
Laplace transforms of impulse functions......Page 440
Relationship between Heaviside step and impulse functions......Page 443
Exercises (25-30)......Page 448
Bending of beams......Page 449
Definitions......Page 453
Stability......Page 456
Impulse response......Page 461
Initial- and final-value theorems......Page 462
Exercises (34-47)......Page 467
Convolution......Page 468
System response to an arbitrary input......Page 471
SISO systems......Page 475
Exercises (53-61)......Page 479
MIMO systems......Page 480
Engineering application: frequency response......Page 487
The pole placement or eigenvalue location technique......Page 495
Exercises (65-70)......Page 497
Review exercises (1–34)......Page 498
The z Transform......Page 506
Introduction......Page 507
Definition and notation......Page 508
Sampling: a first introduction......Page 512
Properties of the z transform......Page 513
The linearity property......Page 514
The first shift property (delaying)......Page 515
The second shift property (advancing)......Page 516
Some further properties......Page 517
Table of z transforms......Page 518
The inverse z transform......Page 519
Inverse techniques......Page 520
Exercises (11-13)......Page 526
Difference equations......Page 527
The solution of difference equations......Page 529
Exercises (14-20)......Page 533
z transfer functions......Page 534
The impulse response......Page 540
Stability......Page 543
Convolution......Page 549
Exercises (21-29)......Page 553
The relationship between Laplace and z transforms......Page 554
State-space model......Page 555
Solution of the discrete-time state equation......Page 558
Exercises (30-33)......Page 562
Euler’s method......Page 563
Step-invariant method......Page 565
Exercises (34-37)......Page 568
Engineering application: design of discrete-time systems......Page 569
Analogue filters......Page 570
Designing a digital replacement filter......Page 571
Introduction......Page 572
The q or shift operator and the δ operator......Page 573
Constructing a discrete-time system model......Page 574
Implementing the design......Page 576
The D transform......Page 578
Review exercises (1–18)......Page 579
Fourier Series......Page 584
Introduction......Page 585
Periodic functions......Page 586
Fourier’s theorem......Page 587
Functions of period 2π......Page 591
Even and odd functions......Page 598
Linearity property......Page 602
Exercises (1-7)......Page 604
Functions of period T......Page 605
Exercises (8-13)......Page 608
Convergence of the Fourier series......Page 609
Full-range series......Page 612
Half-range cosine and sine series......Page 614
Exercises (14-23)......Page 618
Differentiation and integration of Fourier series......Page 619
Integration of a Fourier series......Page 620
Differentiation of a Fourier series......Page 622
Coefficients in terms of jumps at discontinuities......Page 624
Exercises (24-29)......Page 627
Response to periodic input......Page 628
Exercises (30-33)......Page 632
Complex representation......Page 633
The multiplication theorem and Parseval’s theorem......Page 637
Discrete frequency spectra......Page 640
Power spectrum......Page 646
Exercises (34-39)......Page 648
Definitions......Page 649
Generalized Fourier series......Page 651
Convergence of generalized Fourier series......Page 652
Exercises (40-46)......Page 654
Engineering application: describing functions......Page 657
Review exercises (1–20)......Page 658
The Fourier Transform......Page 662
The Fourier integral......Page 663
The Fourier transform pair......Page 669
The continuous Fourier spectra......Page 673
Exercises (1-10)......Page 676
Time-differentiation property......Page 677
Time-shift property......Page 678
Frequency-shift property......Page 679
The symmetry property......Page 680
Exercises (11-16)......Page 682
Relationship between Fourier and Laplace transforms......Page 683
The frequency response......Page 685
Energy and power......Page 688
Convolution......Page 698
Exercises (22-27)......Page 700
A Fourier transform for sequences......Page 701
The discrete Fourier transform......Page 705
Estimation of the continuous Fourier transform......Page 709
The fast Fourier transform......Page 718
Engineering application: the design of analogue filters......Page 725
Introduction......Page 728
Modulation and transmission......Page 730
Identification and isolation of the information-carrying signal......Page 731
Demodulation stage......Page 732
Final signal recovery......Page 733
Digital filters......Page 734
Windows......Page 740
Review exercises (1–25)......Page 744
Partial Differential Equations......Page 748
Introduction......Page 749
Wave equation......Page 750
Heat-conduction or diffusion equation......Page 753
Laplace equation......Page 756
Other and related equations......Page 758
Arbitrary functions and first-order equations......Page 760
Exercises (1-14)......Page 765
D’Alembert solution and characteristics......Page 767
Separated solutions......Page 776
Laplace transform solution......Page 781
Exercises (15-27)......Page 784
Numerical solution......Page 786
Exercises (28-31)......Page 792
Separation method......Page 793
Laplace transform method......Page 797
Exercises (32-40)......Page 802
Numerical solution......Page 804
Separated solutions......Page 810
Exercises (44-54)......Page 818
Numerical solution......Page 819
Exercises (55-59)......Page 826
Finite elements......Page 827
Exercises (60-62)......Page 839
Separated solutions......Page 840
Use of singular solutions......Page 842
Sources and sinks for the heat conduction equation......Page 845
Exercises (63-67)......Page 848
Formal classification......Page 849
Boundary conditions......Page 851
Engineering application: wave propagation under a moving load......Page 856
Engineering application: blood-flow model......Page 859
Review exercises (1–21)......Page 863
Optimization......Page 868
Introduction......Page 869
Introduction......Page 872
Simplex algorithm: an example......Page 874
Simplex algorithm: general theory......Page 878
Exercises (1-11)......Page 885
Two-phase method......Page 886
Exercises (12-20)......Page 894
Equality constraints......Page 895
Inequality constraints......Page 899
Single-variable search......Page 900
Exercises (29-34)......Page 906
Simple multivariable searches......Page 907
Exercises (35-39)......Page 912
Advanced multivariable searches......Page 913
Least squares......Page 917
Exercises (40-43)......Page 920
Engineering application: chemical processing plant......Page 921
Engineering application: heating fin......Page 923
Review exercises (1–26)......Page 926
Applied Probability and Statistics......Page 930
Review of basic probability theory......Page 931
Random variables......Page 932
The Bernoulli, binomial and Poisson distributions......Page 934
The normal distribution......Page 935
Sample measures......Page 936
Interval estimates and hypothesis tests......Page 937
Distribution of the sample average......Page 938
Confidence interval for the mean......Page 939
Testing simple hypotheses......Page 942
Other confidence intervals and tests concerning means......Page 943
Interval and test for proportion......Page 947
Exercises (1-13)......Page 949
Joint distributions and correlation......Page 950
Joint and marginal distributions......Page 951
Independence......Page 953
Covariance and correlation......Page 954
Sample correlation......Page 958
Interval and test for correlation......Page 960
Rank correlation......Page 961
Exercises (14-24)......Page 962
Regression......Page 963
The method of least squares......Page 964
Normal residuals......Page 966
Nonlinear regression......Page 968
Exercises (25-33)......Page 970
Chi-square distribution and test......Page 971
Contingency tables......Page 974
Exercises (34-42)......Page 976
Definition and simple applications......Page 978
The Poisson approximation to the binomial......Page 980
Proof of the central limit theorem......Page 981
Exercises (43-47)......Page 982
Introduction......Page 983
Difference in mean running times and temperatures......Page 984
Dependence of running time on temperature......Page 985
Test for normality......Page 987
Conclusions......Page 988
Shewhart attribute control charts......Page 989
Shewhart variable control charts......Page 992
Cusum control charts......Page 993
Moving-average control charts......Page 996
Exercises (48-59)......Page 998
Typical queueing problems......Page 999
Poisson processes......Page 1000
Single service channel queue......Page 1003
Queues with multiple service channels......Page 1007
Queueing system simulation......Page 1008
Exercises (60-67)......Page 1010
Derivation and simple examples......Page 1011
Applications in probabilistic inference......Page 1013
Exercises (68-78)......Page 1016
Review exercises (1–10)......Page 1017
Answers to Exercises......Page 1020
B......Page 1048
C......Page 1049
D......Page 1050
F......Page 1051
G......Page 1052
I......Page 1053
L......Page 1054
M......Page 1055
O......Page 1056
P......Page 1057
S......Page 1058
U......Page 1060
Z......Page 1061