082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

Recommendations

Info

272 Chapter 8 Matrix algebraTheeffectofpremultiplyingAbyIhasbeentoleaveAunaltered.Theproductisidenticaltothe originalmatrixA, and this iswhyI iscalled an identitymatrix.In general, if A is an arbitrary matrix and I is an identity matrix of the appropriatesize,thenIA=AIfAisasquare matrix thenIA =AI =A.8.5.4 ThetransposeofamatrixIfAisanarbitrarym×nmatrix,arelatedmatrixisthetransposeofA,writtenA T ,foundby interchanging the rows and columns ofA. Thus the first row ofAbecomes the firstcolumn ofA T and so on.A T isann ×mmatrix.( ) 1 −1Example8.15 IfA = findA2 4T .( )Solution 1 2A T =−1 4( ) 426Example8.16 IfA = findA187T and evaluateAA T .⎛ ⎞4 1( ) ⎛ ⎞1Solution A T = ⎝2 8⎠ 426AA T = ⎝42 8⎠ =1876 76 7( ) 56 6262 1148.5.5 SymmetricmatricesIfasquarematrixAanditstransposeA T areidentical,thenAissaidtobeasymmetricmatrix.⎛ ⎞5−4 2Example8.17 IfA = ⎝−4 6 9⎠findA T .2 913⎛ ⎞5−4 2Solution A T = ⎝−4 6 9⎠2 913whichisclearlyequaltoA.HenceAisasymmetricmatrix.Notethatasymmetricmatrixissymmetrical about its leading diagonal.8.5.6 SkewsymmetricmatricesIf a square matrixAissuchthatA T = −A thenAissaidtobe skewsymmetric.
Example8.18 IfA =Solution We haveA T =( ) 0 5, findA−5 0T and deduce thatAisskew symmetric.8.5 Some special matrices 273( ) 0 −5which isclearly equal to −A. HenceAisskew symmetric.5 0EXERCISES8.51 IfA=(a)findA T ,(b)findB T ,(c)findAB,( ) 3 1andB=2 6(d)find (AB) T ,( ) −1 43 8(e)deduce that (AB) T =B T A T .⎛ ⎞x2 Treating the columnvectorx = ⎝y⎠asa3×1zmatrix, findIxwhereI is the 3 ×3 identitymatrix.( ) a b3 IfA= showthatAAc dT is asymmetricmatrix.⎛ ⎞ ⎛ ⎞213 1 −704 IfA= ⎝ 421⎠andB=⎝0 25⎠−132 3 45findA T ,B T ,ABand (AB) T .Deduce that (AB) T =B T A T .5 Determine the type ofmatrixobtained when twodiagonal matrices are multiplied together.Solutions( ) 3 21 (a)1 6( ) 0 20(c)16 56⎛ ⎞x2 ⎝y⎠z4( ) −1 3(b)4 8( ) 0 16(d)20 56⎛ ⎞ ⎛ ⎞24 −1 103⎝12 3⎠,⎝−724⎠,31 2 0556 IfA=( ) a bis skew symmetric, showthatc da =d = 0,that is the diagonal elementsare zero.( ) 1 137 IfA=15 7(a) findA T ,(b) find (A T ) T ,(c) deduce that (A T ) T is equal toA.( ) 9 48 IfA=3 2(a) findA +A T andshowthat thisisasymmetricmatrix,(b) findA −A T andshowthat thisisaskewsymmetric matrix.9 The sumofthe elements onthe leading diagonalofasquarematrixis known asitstrace. Find thetrace of( ) ( )7 2 0 9(a) (b)−1 5 −1 0⎛ ⎞ ⎛ ⎞(c)100⎝010⎠001(d)7 2 1⎝8 2 3⎠9 −1 −4⎛ ⎞ ⎛ ⎞11 020 11 7 5⎝ 7 −20 15⎠,⎝ 0 −20 21⎠5 2125 20 15 255 Diagonal matrix( ) 1 157 (a)13 7( ) 18 78 (a)7 4( ) 1 13(b)15 7( ) 0 1(b)−1 09 (a)12 (b)0 (c)3 (d)5
Page 1 and 2:
ENGINEERINGMATHEMATICSA Foundation
Page 3 and 4:
At Pearson, we have a simple missio
Page 5 and 6:
PEARSONEDUCATIONLIMITEDEdinburghGat
Page 7 and 8:
ContentsPrefaceAcknowledgementsxvii
Page 9 and 10:
Contents ix8.3 Addition,subtraction
Page 11 and 12:
Contents xiChapter17 Numericalinteg
Page 13 and 14:
Contents xiiiChapter24 TheFouriertr
Page 15 and 16:
Contents xvAppendixI Representing a
Page 17 and 18:
PrefaceAudienceThis book has been w
Page 19 and 20:
AcknowledgementsWearegratefultothef
Page 21 and 22:
1 ReviewofalgebraictechniquesConten
Page 23 and 24:
1.2 Laws of indices 3So6 2 6 3 =6 5
Page 25 and 26:
1.2 Laws of indices 5(c)(d)x 9x 5 =
Page 27 and 28:
1.2 Laws of indices 71.2.4 Multiple
Page 29 and 30:
1.2 Laws of indices 9Similarly(2 1/
Page 31 and 32:
1.3 Number bases 11Solutions1 (a) 8
Page 33 and 34:
1.3 Number bases 13giving83=64+16+3
Page 35 and 36:
1.3 Number bases 15Example1.16 Conv
Page 37 and 38:
1.3 Number bases 17Table1.4isusedto
Page 39 and 40:
1.3 Number bases 19The display show
Page 41 and 42:
1.4 Polynomial equations 21Example1
Page 43 and 44:
1.4 Polynomial equations 23We now i
Page 45 and 46:
so thatβ=−13Bycomparing coeffici
Page 47 and 48:
1.5.1 Properandimproperfractions1.5
Page 49 and 50:
1.5 Algebraic fractions 29(b) Facto
Page 51 and 52:
1.5 Algebraic fractions 31and4x +2
Page 53 and 54:
1.6 Solution of inequalities 33(g)(
Page 55 and 56:
1.6 Solution of inequalities 35Exam
Page 57 and 58:
1.6 Solution of inequalities 37Case
Page 59 and 60:
1.7 Partial fractions 391.7 PARTIAL
Page 61 and 62:
1.7 Partial fractions 41Thus we hav
Page 63 and 64:
1.7 Partial fractions 43Solution Th
Page 65 and 66:
1.7 Partial fractions 45EXERCISES1.
Page 67 and 68:
1.8 Summation notation 47Engineerin
Page 69 and 70:
1.8 Summation notation 49thecircuit
Page 71 and 72:
Review exercises 1 513 Removethe br
Page 73 and 74:
Review exercises 1 53[ ((d) 3 z −
Page 75 and 76:
2.2 Numbers and intervals 55in orde
Page 77 and 78:
2.3 Basic concepts of functions 57f
Page 79 and 80:
2.3 Basic concepts of functions 59s
Page 81 and 82:
2.3 Basic concepts of functions 61b
Page 83 and 84:
2.3 Basic concepts of functions 63E
Page 85 and 86:
2.3 Basic concepts of functions 65f
Page 87 and 88:
2.3 Basic concepts of functions 67T
Page 89 and 90:
2.3 Basic concepts of functions 691
Page 91 and 92:
2.4 Review of some common engineeri
Page 93 and 94:
Page 95 and 96:
2.4.2 Rationalfunctions2.4 Review o
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Table2.2Values ofa x fora = 0.5, 2
Page 103 and 104:
Page 105 and 106:
2.4.4 Logarithmfunctions2.4 Review
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Review exercises 2 113REVIEWEXERCIS
Page 135 and 136:
3 ThetrigonometricfunctionsContents
Page 137 and 138:
Note thattanθ = BCAB = BCAC × ACA
Page 139 and 140:
3.3 The trigonometric ratios 119yyB
Page 141 and 142:
3.4 The sine, cosine and tangent fu
Page 143 and 144:
3.5 The sincxfunction 123EXERCISES3
Page 145 and 146:
3.6 Trigonometric identities 125ors
Page 147 and 148:
3.6 Trigonometric identities 127Exa
Page 149 and 150:
3.6 Trigonometric identities 129Exa
Page 151 and 152:
3.7 Modelling waves using sint and
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
3.8 Trigonometric equations 145sin
Page 167 and 168:
3.8 Trigonometric equations 147tan
Page 169 and 170:
Usingascientific calculator wehavez
Page 171 and 172:
Review exercises 3 1516 Simplify th
Page 173 and 174:
Review exercises 3 153(d) 1.581sin(
Page 175 and 176:
4.2 Cartesian coordinate system --
Page 177 and 178:
4.3 Cartesian coordinate system --
Page 179 and 180:
4.4 Polar coordinates 1594.4 POLARC
Page 181 and 182:
4.4 Polar coordinates 161yOurxFigur
Page 183 and 184:
4.5 Some simple polar curves 163EXE
Page 185 and 186:
4.5 Some simple polar curves 165uAn
Page 187 and 188:
4.6 Cylindrical polar coordinates 1
Page 189 and 190:
4.6 Cylindrical polar coordinates 1
Page 191 and 192:
4.7 Spherical polar coordinates 171
Page 193 and 194:
Review exercises 4 173zzFeeding lin
Page 195 and 196:
5 DiscretemathematicsContents 5.1 I
Page 197 and 198:
5.2 Set theory 177Example5.1 Useset
Page 199 and 200:
5.2 Set theory 179EMNM fNfFigure5.2
Page 201 and 202:
5.2 Set theory 181someelementsinA.W
Page 203 and 204:
5.3 Logic 183CDCD(a)(b)CDCD(c)(d)(e
Page 205 and 206:
5.4 Boolean algebra 185ABA . BFigur
Page 207 and 208:
5.4 Boolean algebra 187Table5.12Tru
Page 209 and 210:
5.4 Boolean algebra 189However, we
Page 211 and 212:
5.4 Boolean algebra 191S0Stage0A0 B
Page 213 and 214:
5.4 Boolean algebra 193V DDPMOSQ1DD
Page 215 and 216:
5.4 Boolean algebra 195(d) A ·B·(
Page 217 and 218:
Review exercises 5 19712 (a) A·B+A
Page 219 and 220:
Review exercises 5 1997 (a) (A·B)
Page 221 and 222:
6.2 Sequences 2016.2 SEQUENCESAsequ
Page 223 and 224:
6.2 Sequences 203sin t1x[k]1- 3p-2p
Page 225 and 226:
6.2 Sequences 205Ageneralgeometricp
Page 227 and 228:
6.2 Sequences 207can sensibly letk
Page 229 and 230:
6.3 Series 20913 ±12814 315 016 (a
Page 231 and 232:
6.3.2 Sumofafinitegeometricseries6.
Page 233 and 234:
6.3 Series 213whereS ∞is known as
Page 235 and 236:
6.4 The binomial theorem 215Example
Page 237 and 238:
6.4 The binomial theorem 217EXERCIS
Page 239 and 240:
6.6 Sequences arising from the iter
Page 241 and 242: 6.6 Sequences arising from the iter
Page 243 and 244: Review exercises 6 22315 (a) Write
Page 245 and 246: 7.2 Vectors and scalars: basic conc
Page 253 and 254: yjOirFigure7.19Thex--y plane withpo
Page 255 and 256: 7.3 Cartesian components 235Solutio
Page 257 and 258: 7.3 Cartesian components 237Enginee
Page 259 and 260: 7.3 Cartesian components 2393 Ifa=4
Page 261 and 262: 7.5 The scalar product 241OEFigure7
Page 263 and 264: 7.5 The scalar product 243Example7.
Page 265 and 266: ----------7.5 The scalar product 24
Page 267 and 268: 7.6 The vector product 247a 3 bubPl
Page 269 and 270: 7.6 The vector product 249The k com
Page 271 and 272: 7.6 The vector product 251Engineeri
Page 273 and 274: 7.7 Vectors ofndimensions 253andH =
Page 275 and 276: Review exercises 7 255EXERCISES7.71
Page 277 and 278: 8 MatrixalgebraContents 8.1 Introdu
Page 279 and 280: 8.3 Addition, subtraction and multi
Page 283 and 284: Example8.8 IfB =( ) 123andC=456⎛
Page 287 and 288: 8.4 Using matrices in the translati
Page 289 and 290: 8.4 Using matrices in the translati
Page 291: 8.5 Some special matrices 271V new=
Page 295 and 296: 8.6.1 FindingtheinverseofamatrixFor
Page 297 and 298: 8.6 The inverse of a 2 × 2 matrix
Page 299 and 300: 8.7 Determinants 279In addition to
Page 301 and 302: 8.8 The inverse of a 3 × 3 matrix
Page 303 and 304: 8.9 Application to the solution of
Page 305 and 306: 8.9 Application to the solution of
Page 307 and 308: Solution Firstconsider the equation
Page 309 and 310: 8.10 Gaussian elimination 289anythi
Page 311 and 312: Eliminating the unwanted values int
Page 313 and 314: ⎛R 1R 2→R 2−6R 1⎝R 3→R 3
Page 315 and 316: 8.11 Eigenvalues and eigenvectors 2
Page 319 and 320: Itfollows thatso that(2−λ)(4−
Page 327 and 328: 8.12 Analysis of electrical network
Page 333 and 334: 8.13 Iterative techniques for the s
Page 339 and 340: 8.14 Computer solutions of matrix p
Page 341 and 342: Review exercises 8 321V = [3;−6;4
Page 343 and 344:
Review exercises 8 3234 (a) −8⎛
Page 345 and 346:
9.2 Complex numbers 3259.2 COMPLEXN
Page 347 and 348:
9.2 Complex numbers 3279.2.1 Thecom
Page 349 and 350:
Similarly, wefind, by equating imag
Page 351 and 352:
9.3 Operations with complex numbers
Page 353 and 354:
9.5 Polar form of a complex number
Page 355 and 356:
9.5.1 Multiplicationanddivisioninpo
Page 357 and 358:
9.7 The exponential form of a compl
Page 359 and 360:
9.7 The exponential form of a compl
Page 361 and 362:
9.8 Phasors 341example, in the case
Page 363 and 364:
9.8 Phasors 343Therefore,Ṽ S=ṼR
Page 365 and 366:
which isclearly true,and the theore
Page 367 and 368:
9.9 De Moivre’s theorem 347y25p
Page 369 and 370:
9.9 De Moivre’s theorem 349Now, w
Page 371 and 372:
9.10 Loci and regions of the comple
Page 373 and 374:
9.10 Loci and regions of the comple
Page 375 and 376:
Review exercises 9 3559 Sketch the
Page 377 and 378:
10.2 Graphical approach to differen
Page 379 and 380:
10.3 Limits and continuity 359f (t)
Page 381 and 382:
10.3 Limits and continuity 361Theli
Page 383 and 384:
10.4 Rate of change at a specific p
Page 385 and 386:
10.5 Rate of change at a general po
Page 387 and 388:
Page 389 and 390:
Page 391 and 392:
10.6 Existence of derivatives 371x
Page 393 and 394:
10.7 Common derivatives 373Table10.
Page 395 and 396:
10.8 Differentiation as a linear op
Page 397 and 398:
Page 399 and 400:
Page 401 and 402:
Page 403 and 404:
Page 405 and 406:
Review exercises 10 385REVIEWEXERCI
Page 407 and 408:
11.2 Rules of differentiation 387Th
Page 409 and 410:
11.2 Rules of differentiation 389Th
Page 411 and 412:
(b) Ify = ln(1 −t),theny ′ =
Page 413 and 414:
11.3 Parametric, implicit and logar
Page 415 and 416:
Page 417 and 418:
Page 419 and 420:
Page 421 and 422:
11.4 Higher derivatives 401Example1
Page 423 and 424:
11.4 Higher derivatives 403yCByCAyB
Page 425 and 426:
Review exercises 11 405Solutions1 (
Page 427 and 428:
12.2 Maximum points and minimum poi
Page 429 and 430:
Page 431 and 432:
Page 433 and 434:
Page 435 and 436:
12.3 Points of inflexion 415Solutio
Page 437 and 438:
12.3 Points of inflexion 417y ′ =
Page 439 and 440:
12.4 The Newton--Raphson method for
Page 441 and 442:
12.4 The Newton--Raphson method for
Page 443 and 444:
12.5 Differentiation of vectors 423
Page 445 and 446:
Solution (a) Ifa = 3t 2 i +cos2tj,
Page 447 and 448:
Page 449 and 450:
13.2 Elementary integration 42913.2
Page 451 and 452:
13.2 Elementary integration 431Tabl
Page 453 and 454:
Solution (a)sinx+cosx(j)2(k) 2t −
Page 455 and 456:
13.2 Elementary integration 435Engi
Page 457 and 458:
13.2 Elementary integration 4374.00
Page 459 and 460:
Therefore,∫∫ 1sinmtsinnt dt = {
Page 461 and 462:
13.2 Elementary integration 4412 (a
Page 463 and 464:
13.3 Definite and indefinite integr
Page 465 and 466:
Page 467 and 468:
Page 469 and 470:
Page 471 and 472:
Page 473 and 474:
Review exercises 13 4536716438 (a)
Page 475 and 476:
Review exercises 13 45522 Calculate
Page 477 and 478:
14 TechniquesofintegrationContents
Page 479 and 480:
14.2 Integration by parts 459Usingt
Page 481 and 482:
Usingintegration by parts we have
Page 483 and 484:
14.3 Integration by substitution 46
Page 485 and 486:
14.3 Integration by substitution 46
Page 487 and 488:
14.4 Integration using partial frac
Page 489 and 490:
Review exercises 14 469∫ π/3(b)
Page 491 and 492:
15 ApplicationsofintegrationContent
Page 493 and 494:
15.2 Average value of a function 47
Page 495 and 496:
15.3 Root mean square value of a fu
Page 497 and 498:
15.3 Root mean square value of a fu
Page 499 and 500:
Review exercises 15 479(d) 0.7071(e
Page 501 and 502:
[Example16.1 Show that f (x) =xandg
Page 503 and 504:
Solution (a) We use the trigonometr
Page 505 and 506:
16.3 Improper integrals 485iC 1 C 2
Page 507 and 508:
∫ 2either of the integrals failst
Page 509 and 510:
16.4 Integral properties of the del
Page 511 and 512:
16.5 Integration of piecewise conti
Page 513 and 514:
16.6 Integration of vectors 49316.6
Page 515 and 516:
Review exercises 16 49512 Given14 G
Page 517 and 518:
17.2 Trapezium rule 497y{ {y 0 y ny
Page 519 and 520:
17.2 Trapezium rule 499The differen
Page 521 and 522:
yABCDE17.3 Simpson’s rule 501quad
Page 523 and 524:
17.3 Simpson’s rule 503Table17.6T
Page 525 and 526:
Review Exercises 17 505EXERCISES17.
Page 527 and 528:
18 Taylorpolynomials,Taylorseriesan
Page 529 and 530:
18.2 Linearization using first-orde
Page 531 and 532:
18.2 Linearization using first-orde
Page 533 and 534:
18.3 Second-order Taylor polynomial
Page 535 and 536:
18.3 Second-order Taylor polynomial
Page 537 and 538:
18.4 Taylor polynomials of the nth
Page 539 and 540:
18.4. Taylor polynomials of the nth
Page 541 and 542:
18.5 Taylor’s formula and the rem
Page 543 and 544:
We know |x| < 1, and so 4x51518.5 T
Page 545 and 546:
18.6 Taylor and Maclaurin series 52
Page 547 and 548:
Page 549 and 550:
Page 551 and 552:
Page 553 and 554:
Review exercises 18 5337 (a) 0, 2,
Page 555 and 556:
19.2 Basic definitions 535In engine
Page 557 and 558:
19.2 Basic definitions 537Note also
Page 559 and 560:
19.2 Basic definitions 539fromwhich
Page 561 and 562:
19.3 First-order equations: simple
Page 563 and 564:
Page 565 and 566:
Page 567 and 568:
19.4 First-order linear equations:
Page 569 and 570:
and so, upon integrating,so that(co
Page 571 and 572:
Page 573 and 574:
Page 575 and 576:
Page 577 and 578:
Page 579 and 580:
19.5 Second-order linear equations
Page 581 and 582:
19.6 Constant coefficient equations
Page 583 and 584:
Page 585 and 586:
Page 587 and 588:
Page 589 and 590:
Page 591 and 592:
thatis,β = 332The required particu
Page 593 and 594:
Page 595 and 596:
Page 597 and 598:
Page 599 and 600:
Page 601 and 602:
Page 603 and 604:
Page 605 and 606:
19.7 Series solution of differentia
Page 607 and 608:
19.8 Bessel’s equation and Bessel
Page 609 and 610:
Page 611 and 612:
fromwhicha 5=− 124 a 3 = 1192 a 1
Page 613 and 614:
n 5 2n 5 3n5 419.8 Bessel’s equat
Page 615 and 616:
Page 617 and 618:
Page 619 and 620:
Page 621 and 622:
Review exercises 19 601quencies, kn
Page 623 and 624:
20 OrdinarydifferentialequationsIIC
Page 625 and 626:
20.2 Analogue simulation 605y iy i
Page 627 and 628:
20.3 Higher order equations 607High
Page 629 and 630:
20.4 State-space models 609Solution
Page 631 and 632:
20.4 State-space models 611statevar
Page 633 and 634:
20.4 State-space models 613R a L aT
Page 635 and 636:
20.5 Numerical methods 615Combining
Page 637 and 638:
20.6 Euler’s method 617and we can
Page 639 and 640:
20.6 Euler’s method 619Whenx = 1,
Page 641 and 642:
20.7 Improved Euler method 621Table
Page 643 and 644:
20.8 Runge--Kutta method of order 4
Page 645 and 646:
20.8.1 Higherorderequations20.8 Run
Page 647 and 648:
21 TheLaplacetransformContents 21.1
Page 649 and 650:
21.3 Laplace transforms of some com
Page 651 and 652:
21.4 Properties of the Laplace tran
Page 653 and 654:
(b) Usethe first shift theorem with
Page 655 and 656:
21.5 Laplace transform of derivativ
Page 657 and 658:
21.5 Laplace transform of derivativ
Page 659 and 660:
21.6 Inverse Laplace transforms 639
Page 661 and 662:
21.7 Using partial fractions to fin
Page 663 and 664:
21.8 Finding the inverse Laplace tr
Page 665 and 666:
21.8 Finding the inverse Laplace tr
Page 667 and 668:
21.9 The convolution theorem 64721.
Page 669 and 670:
21.9 Solving linear constant coeffi
Page 671 and 672:
21.10 Solving linear constant coeff
Page 673 and 674:
Page 675 and 676:
X(s)= s3 −3s 2 −4s+6s 2 (s 2
Page 677 and 678:
Page 679 and 680:
21.11 Transfer functions 659(c) x
Page 681 and 682:
21.11 Transfer functions 661R(s)G(s
Page 683 and 684:
21.11 Transfer functions 663R(s) +-
Page 685 and 686:
21.11 Transfer functions 665Fuel va
Page 687 and 688:
21.11 Transfer functions 667u a (t)
Page 689 and 690:
21.12 Poles, zeros and thesplane 66
Page 691 and 692:
Page 693 and 694:
Page 695 and 696:
21.13 Laplace transforms of some sp
Page 697 and 698:
21.13.2 Periodicfunctions21.13 Lapl
Page 699 and 700:
Review exercises 21 6792 Find the L
Page 701 and 702:
22 Differenceequationsandtheztransf
Page 703 and 704:
22.2 Basic definitions 68322.2.2 Th
Page 705 and 706:
22.2 Basic definitions 685an inhomo
Page 707 and 708:
22.3 Rewriting difference equations
Page 709 and 710:
22.4 Block diagram representation o
Page 711 and 712:
22.4 Block diagram representation o
Page 713 and 714:
22.5 Design of a discrete-time cont
Page 715 and 716:
22.6 Numerical solution of differen
Page 717 and 718:
22.6 Numerical solution of differen
Page 719 and 720:
22.7 Definition of the z transform
Page 721 and 722:
22.7 Definition of the z transform
Page 723 and 724:
22.8 Sampling a continuous signal 7
Page 725 and 726:
22.9 The relationship between the z
Page 727 and 728:
22.9 The relationship between the z
Page 729 and 730:
22.10 Properties of the z transform
Page 731 and 732:
Page 733 and 734:
Page 735 and 736:
22.11 Inversion of z transforms 715
Page 737 and 738:
22.11 Inversion of z transforms 717
Page 739 and 740:
22.12 The z transform and differenc
Page 741 and 742:
Review exercises 22 721(e) q[k +3]
Page 743 and 744:
23.2 Periodic waveforms 72323.2 PER
Page 745 and 746:
23.2 Periodic waveforms 725f(t)1f(t
Page 747 and 748:
23.3 Odd and even functions 727f(t)
Page 749 and 750:
23.3 Odd and even functions 729f(t)
Page 751 and 752:
23.3 Odd and even functions 731the
Page 753 and 754:
23.5 Fourier series 733Table23.1Som
Page 755 and 756:
23.5 Fourier series 735point out th
Page 757 and 758:
23.5 Fourier series 737but cos(−n
Page 759 and 760:
23.5 Fourier series 739Example23.16
Page 761 and 762:
23.5 Fourier series 741y4- 3π -- -
Page 763 and 764:
23.5 Fourier series 743We conclude
Page 765 and 766:
23.6 Half-range series 745EXERCISES
Page 767 and 768:
23.6 Half-range series 747Half-rang
Page 769 and 770:
23.8 Complex notation 749Engineerin
Page 771 and 772:
23.9 Frequency response of a linear
Page 773 and 774:
23.9 Frequency response of a linear
Page 775 and 776:
̸̸̸Review exercises 23 755The va
Page 777 and 778:
24 TheFouriertransformContents 24.1
Page 779 and 780:
24.2 The Fourier transform -- defin
Page 781 and 782:
24.3 Some properties of the Fourier
Page 783 and 784:
∫ 2[ ] eSolution (a) F(ω)=3 e
Page 785 and 786:
24.3 Some properties of the Fourier
Page 787 and 788:
24.4 Spectra 767uF(v)u2-4p -3p -2p
Page 789 and 790:
The t−ω duality principle 769f(t
Page 791 and 792:
24.6 Fourier transforms of some spe
Page 793 and 794:
24.7 The relationship between the F
Page 795 and 796:
24.8 Convolution and correlation 77
Page 797 and 798:
and theirproduct isF(ω)G(ω) =1(1+
Page 799 and 800:
Page 801 and 802:
Page 803 and 804:
24.9 The discrete Fourier transform
Page 805 and 806:
24.9 The discrete Fourier transform
Page 807 and 808:
24.10 Derivation of the d.f.t. 787T
Page 809 and 810:
(Solution ConsiderF˜ ω + 2π )for
Page 811 and 812:
24.11 Using the d.f.t.to estimate a
Page 813 and 814:
24.13 Some properties of the d.f.t.
Page 815 and 816:
24.14 The discrete cosine transform
Page 817 and 818:
Page 819 and 820:
Page 821 and 822:
24.15 Discrete convolution and corr
Page 823 and 824:
Page 825 and 826:
Page 827 and 828:
Page 829 and 830:
Page 831 and 832:
Page 833 and 834:
Page 835 and 836:
Page 837 and 838:
Page 839 and 840:
Page 841 and 842:
Page 843 and 844:
25 FunctionsofseveralvariablesConte
Page 845 and 846:
25.3 Partial derivatives 825DCAz =
Page 847 and 848:
25.3 Partial derivatives 827Enginee
Page 849 and 850:
25.4 Higher order derivatives 829
Page 851 and 852:
25.4 Higher order derivatives 831Ex
Page 853 and 854:
25.5 Partial differential equations
Page 855 and 856:
25.6 Taylor polynomials and Taylor
Page 857 and 858:
Page 859 and 860:
Page 861 and 862:
25.7 Maximum and minimum points of
Page 863 and 864:
Page 865 and 866:
Page 867 and 868:
Review exercises 25 8474 Find allfi
Page 869 and 870:
26 VectorcalculusContents 26.1 Intr
Page 871 and 872:
26.3 The gradient of a scalar field
Page 873 and 874:
(b) At (0,0,0), ∇φ =0i+3j+0k =3j
Page 875 and 876:
26.3 The gradient of a scalar field
Page 877 and 878:
26.4.1 Physicalinterpretationof ∇
Page 879 and 880:
26.5 The curl of a vector field 859
Page 881 and 882:
26.6 Combining the operators grad,
Page 883 and 884:
26.6 Combining the operators grad,
Page 885 and 886:
Equation3Review exercises 26 865cur
Page 887 and 888:
27 Lineintegralsandmultipleintegral
Page 889 and 890:
27.2 Line integrals 869SolutionThe
Page 891 and 892:
27.3 Evaluation of line integrals i
Page 893 and 894:
27.4 Evaluation of line integrals i
Page 895 and 896:
27.5 Conservative fields and potent
Page 897 and 898:
27.5 Conservative fields and potent
Page 899 and 900:
Substitutingy =x 2 and dy = 2xdx we
Page 901 and 902:
27.6 Double and triple integrals 88
Page 903 and 904:
Page 905 and 906:
Page 907 and 908:
y10GDR27.6 Double and triple integr
Page 909 and 910:
27.7 Some simple volume and surface
Page 911 and 912:
Page 913 and 914:
Page 915 and 916:
27.8 The divergence theorem and Sto
Page 917 and 918:
27.8 The divergence theorem and Sto
Page 919 and 920:
27.9 Maxwell’s equations in integ
Page 921 and 922:
Page 923 and 924:
28 ProbabilityContents 28.1 Introdu
Page 925 and 926:
28.2 Introducing probability 905Ifm
Page 927 and 928:
28.2 Introducing probability 907Eng
Page 929 and 930:
28.3 Mutually exclusive events: the
Page 931 and 932:
28.3 Mutually exclusive events: the
Page 933 and 934:
28.4 Complementary events 913EXERCI
Page 935 and 936:
28.5 Concepts from communication th
Page 937 and 938:
28.5 Concepts from communication th
Page 939 and 940:
28.6 Conditional probability: the m
Page 941 and 942:
28.6 Conditional probability: the m
Page 943 and 944:
P(A ∩C) =P(C ∩A) =P(C)P(A|C)P(A
Page 945 and 946:
28.7 Independent events 9255 Compon
Page 947 and 948:
28.7 Independent events 927Solution
Page 949 and 950:
28.7 Independent events 929E 1,E 2a
Page 951 and 952:
Review exercises 28 931(d) A compon
Page 953 and 954:
29 Statisticsandprobabilitydistribu
Page 955 and 956:
29.3 Probability distributions -- d
Page 957 and 958:
29.4 Probability density functions
Page 959 and 960:
29.5 Mean value 939Example29.3 Find
Page 961 and 962:
29.6 Standard deviation 94129.6 STA
Page 963 and 964:
29.7 Expected value of a random var
Page 965 and 966:
29.7 Expected value of a random var
Page 967 and 968:
29.8 Standard deviation of a random
Page 969 and 970:
29.9 Permutations and combinations
Page 971 and 972:
29.9 Permutations and combinations
Page 973 and 974:
29.10 The binomial distribution 953
Page 975 and 976:
29.10 The binomial distribution 955
Page 977 and 978:
29.11 The Poisson distribution 957I
Page 979 and 980:
Table29.6Theprobabilitiesforbinomia
Page 981 and 982:
29.12 The uniform distribution 9615
Page 983 and 984:
29.14 The normal distribution 963So
Page 985 and 986:
29.14 The normal distribution 965N(
Page 987 and 988:
29.14 The normal distribution 967Ta
Page 989 and 990:
29.14 The normal distribution 969EX
Page 991 and 992:
29.15 Reliability engineering 971in
Page 993 and 994:
29.15 Reliability engineering 973En
Page 995 and 996:
29.15 Reliability engineering 975k
Page 997 and 998:
Review exercises 29 977Solutions1 0
Page 999 and 1000:
AppendicesContents I Representingac
Page 1001 and 1002:
II The Greek alphabet 981f(t)T0 t 0
Page 1003 and 1004:
Indexabsolute quantity88absorption
Page 1005 and 1006:
Index 985chord 357--8circuital law
Page 1007 and 1008:
Index 987cycle ofsint 131cycles ofl
Page 1009 and 1010:
Index 989eigenvalues andeigenvector
Page 1011 and 1012:
Index 991spectra 766--8t-w duality
Page 1013 and 1014:
Index 993characteristic impedance o
Page 1015 and 1016:
Index 995exclusive OR gate 189NAND
Page 1017 and 1018:
Index 997second-order 829--30, 833,
Page 1019 and 1020:
Index 999remainder term,Taylor’s
Page 1021 and 1022:
Index 1001solenoidal vectorfield 85
Page 1023 and 1024:
Index 1003calculus 849--66curl859--
show all

082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?