Instructors' Solutions for Mathematical Methods for Physics and Engineering (third edition) PDF

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Instructors’ Solutions for Mathematical Methods for Physics and Engineering (third edition) K.F. Riley and M.P. Hobson Contents Introduction xvii 1 Preliminary algebra 1 1.2 1 1.4 1 1.6 2 1.8 3 1.10 4 1.12 4 1.14 5 1.16 5 1.18 6 1.20 8 1.22 8 1.24 9 1.26 10 1.28 11 1.30 12 1.32 13 2 Preliminary calc...

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Instructors’ Solutions for Mathematical Methods for Physics and Engineering (third edition) K.F. Riley and M.P. Hobson

Contents

xvii

Introduction 1

2

Preliminary algebra

1

1.2

1

1.4

1

1.6

2

1.8

3

1.10

4

1.12

4

1.14

5

1.16

5

1.18

6

1.20

8

1.22

8

1.24

9

1.26

10

1.28

11

1.30

12

1.32

13

Preliminary calculus

15

2.2

15 iii

CONTENTS

3

2.4

15

2.6

16

2.8

17

2.10

17

2.12

19

2.14

20

2.16

22

2.18

23

2.20

24

2.22

25

2.24

25

2.26

26

2.28

27

2.30

28

2.32

29

2.34

30

2.36

31

2.38

33

2.40

34

2.42

35

2.44

36

2.46

37

2.48

39

2.50

39

Complex numbers and hyperbolic functions

43

3.2

43

3.4

44

3.6

45

3.8

46

3.10

47

3.12

49 iv

CONTENTS

4

5

3.14

50

3.16

51

3.18

52

3.20

53

3.22

53

3.24

54

3.26

56

3.28

57

Series and limits

58

4.2

58

4.4

58

4.6

59

4.8

59

4.10

61

4.12

62

4.14

62

4.16

63

4.18

63

4.20

64

4.22

66

4.24

67

4.26

69

4.28

70

4.30

72

4.32

72

4.34

73

4.36

74

Partial differentiation

75

5.2

75

5.4

76 v

CONTENTS

6

7

5.6

77

5.8

78

5.10

79

5.12

80

5.14

81

5.16

82

5.18

82

5.20

83

5.22

84

5.24

86

5.26

87

5.28

89

5.30

90

5.32

91

5.34

92

Multiple integrals

93

6.2

93

6.4

93

6.6

95

6.8

95

6.10

96

6.12

97

6.14

98

6.16

99

6.18

100

6.20

101

6.22

103

Vector algebra

105

7.2

105

7.4

105 vi

CONTENTS

8

7.6

106

7.8

106

7.10

107

7.12

108

7.14

108

7.16

110

7.18

110

7.20

111

7.22

112

7.24

114

7.26

115

Matrices and vector spaces

117

8.2

117

8.4

118

8.6

120

8.8

122

8.10

122

8.12

123

8.14

125

8.16

126

8.18

127

8.20

128

8.22

130

8.24

131

8.26

131

8.28

132

8.30

133

8.32

134

8.34

135

8.36

136

8.38

137 vii

CONTENTS

9

10

11

8.40

139

8.42

140

Normal modes

144

9.2

144

9.4

146

9.6

148

9.8

149

9.10

151

Vector calculus

153

10.2

153

10.4

154

10.6

155

10.8

156

10.10

157

10.12

158

10.14

159

10.16

161

10.18

161

10.20

164

10.22

165

10.24

167

Line, surface and volume integrals

170

11.2

170

11.4

171

11.6

172

11.8

173

11.10

174

11.12

175

11.14

176

11.16

177 viii

CONTENTS

12

13

11.18

178

11.20

179

11.22

180

11.24

181

11.26

183

11.28

184

Fourier series

186

12.2

186

12.4

186

12.6

187

12.8

189

12.10

190

12.12

191

12.14

192

12.16

193

12.18

194

12.20

195

12.22

197

12.24

198

12.26

199

Integral transforms

202

13.2

202

13.4

203

13.6

205

13.8

206

13.10

208

13.12

210

13.14

211

13.16

211

13.18

213 ix

CONTENTS

14

15

13.20

214

13.22

216

13.24

217

13.26

219

13.28

220

First-order ODEs

223

14.2

223

14.4

224

14.6

224

14.8

225

14.10

226

14.12

227

14.14

228

14.16

228

14.18

229

14.20

230

14.22

232

14.24

233

14.26

234

14.28

235

14.30

236

Higher-order ODEs

237

15.2

237

15.4

238

15.6

240

15.8

241

15.10

242

15.12

243

15.14

245

15.16

247 x

CONTENTS

16

17

18

15.18

248

15.20

249

15.22

250

15.24

251

15.26

253

15.28

254

15.30

255

15.32

256

15.34

258

15.36

259

Series solutions of ODEs

261

16.2

261

16.4

262

16.6

264

16.8

266

16.10

268

16.12

270

16.14

271

16.16

272

Eigenfunction methods for ODEs

274

17.2

274

17.4

276

17.6

277

17.8

279

17.10

280

17.12

282

17.14

284

Special functions

285

18.2

285

18.4

286 xi

CONTENTS

19

20

21

18.6

287

18.8

288

18.10

290

18.12

291

18.14

293

18.16

294

18.18

295

18.20

297

18.22

298

18.24

300

Quantum operators

303

19.2

303

19.4

304

19.6

305

19.8

308

19.10

309

PDEs; general and particular solutions

312

20.2

312

20.4

313

20.6

315

20.8

316

20.10

317

20.12

318

20.14

318

20.16

319

20.18

321

20.20

322

20.22

323

20.24

324

PDEs: separation of variables

326 xii

CONTENTS

22

23

21.2

326

21.4

328

21.6

329

21.8

331

21.10

332

21.12

334

21.14

336

21.16

336

21.18

338

21.20

339

21.22

341

21.24

343

21.26

344

21.28

346

Calculus of variations

348

22.2

348

22.4

349

22.6

350

22.8

351

22.10

352

22.12

353

22.14

354

22.16

355

22.18

355

22.20

356

22.22

357

22.24

359

22.26

361

22.28

363

Integral equations

366 xiii

CONTENTS

24

25

23.2

366

23.4

366

23.6

368

23.8

370

23.10

371

23.12

372

23.14

373

23.16

374

Complex variables

377

24.2

377

24.4

378

24.6

379

24.8

380

24.10

381

24.12

383

24.14

384

24.16

385

24.18

386

24.20

387

24.22

388

Applications of complex variables

390

25.2

390

25.4

391

25.6

393

25.8

394

25.10

396

25.12

398

25.14

399

25.16

401

25.18

402 xiv

CONTENTS

26

27

25.20

404

25.22

406

Tensors

409

26.2

409

26.4

410

26.6

411

26.8

413

26.10

414

26.12

415

26.14

417

26.16

418

26.18

419

26.20

420

26.22

421

26.24

422

26.26

423

26.28

426

Numerical methods

428

27.2

428

27.4

428

27.6

429

27.8

431

27.10

432

27.12

433

27.14

435

27.16

436

27.18

438

27.20

440

27.22

441

27.24

442 xv

CONTENTS

28

29

30

27.26

444

Group theory

447

28.2

447

28.4

448

28.6

449

28.8

450

28.10

452

28.12

453

28.14

455

28.16

456

28.18

457

28.20

458

28.22

460

Representation theory

462

29.2

462

29.4

464

29.6

467

29.8

470

29.10

472

29.12

475

Probability

479

30.2

479

30.4

480

30.6

481

30.8

483

30.10

484

30.12

485

30.14

486

30.16

487

30.18

489 xvi

CONTENTS

31

30.20

490

30.22

491

30.24

494

30.26

494

30.28

496

30.30

497

30.32

498

30.34

499

30.36

501

30.38

502

30.40

503

Statistics

505

31.2

505

31.4

506

31.6

507

31.8

508

31.10

511

31.12

513

31.14

514

31.16

516

31.18

517

31.20

518

xvii

Introduction

The second edition of Mathematical Methods for Physics and Engineering carried more than twice as many exercises, based on its various chapters, as did the first. In the Preface we discussed the general question of how such exercises should be treated but, in the end, decided to provide hints and outline answers to all problems, as in the first edition. This decision was an uneasy one as, on the one hand, it did not allow the exercises to be set as totally unaided homework that could be used for assessment purposes but, on the other, it did not give a full explanation of how to tackle a problem when a student needed explicit guidance or a model answer. In order to allow both of these educationally desirable goals to be achieved we have, in the third edition, completely changed the way this matter is handled. All of the exercises from the second edition, plus a number of additional ones testing the newly-added material, have been included in penultimate subsections of the appropriate, sometimes reorganised, chapters. Hints and outline answers are given, as previously, in the final subsections, but only to the odd-numbered exercises. This leaves all even-numbered exercises free to be set as unaided homework, as described below. For the four hundred plus odd-numbered exercises, complete solutions are available, to both students and their teachers, in the form of a separate manual, K. F. Riley and M. P. Hobson, Student Solutions Manual for Mathematical Methods for Physics and Engineering, 3rd edn. (Cambridge: CUP, 2006). These full solutions are additional to the hints and outline answers given in the main text. For each exercise, the original question is reproduced and then followed by a fully-worked solution. For those exercises that make internal reference to the main text or to other (even-numbered) exercises not included in the manual, the questions have been reworded, usually by including additional information, so that the questions can stand alone. xix

INTRODUCTION

The remaining four hundred or so even-numbered exercises have no hints or answers, outlined or detailed, available for general access. They can therefore be used by instructors as a basis for setting unaided homework. Full solutions to these exercises, in the same general format as those appearing in the manual (though they may contain cross-references to the main text or to other exercises), form the body of the material on this website. In many cases, in the manual as well as here, the solution given is even fuller than one that might be expected of a good student who has understood the material. This is because we have aimed to make the solutions instructional as well as utilitarian. To this end, we have included comments that are intended to show how the plan for the solution is fomulated and have given the justifications for particular intermediate steps (something not always done, even by the best of students). We have also tried to write each individual substituted formula in the form that best indicates how it was obtained, before simplifying it at the next or a subsequent stage. Where several lines of algebraic manipulation or calculus are needed to obtain a final result they are normally included in full; this should enable the instructor to determine whether a student’s incorrect answer is due to a misunderstanding of principles or to a technical error. In all new publications, on paper or on a website, errors and typographical mistakes are virtually unavoidable and we would be grateful to any instructor who brings instances to our attention. Ken Riley, [email protected], Michael Hobson, [email protected], Cambridge, 2006

xx