AdvancedEngineeringMathematics

A new edition of FurtherEngineering Mathematics

K. A . StroudFormerly Principal Lecturer

Department of Mathematics, Coventry University

with additions by

Dexter j. BoothPrincipal Lecturer

School of Computing and Engineering, University of Huddersfield

FOURTH EDITION

Review Board for the fourth edition :Dr Mike Gover, University of BradfordDr Pat Lewis, Staffordshire UniversityDr Phil Everson, University of Exeter

Dr Marc Andre Armand, National University of SingaporeDr Lilla Ferrarlo, The Australian National University

Dr Bernadine Renaldo Wong, University of Malaya, Malaysia

Additional reviewers :Dr John Appleby, University of NewcastleDr John Dormand, University of Teesside

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Contents

Preface to the First Edition

xvPreface to the Second Edition

xviiPreface to the Third Edition

xviiiPreface to the Fourth Edition

xixHints on using the book

xxiUseful background information

xxii

Programme 1_Numerical solutions ofequations and interpolation

Learning outcomes

1Introduction

2The Fundamental Theorem of Algebra

2Relations between the coefficients and the roots of a

polynomial equation

4Cubic equations

7Transforming a cubic to reduced form

7Tartaglia's solution for a real root

8Numerical methods

9Bisection

9Numerical solution of equations by iteration

11Using a spreadsheet

12Relative addresses

13Newton--Raphson iterative method

14Tabular display of results

16Modified Newton-Raphson method

21Interpolation

24Linear interpolation

24Graphical interpolation

25Gregory--Newton interpolation formula using forward finite

differences

25Central differences

31GregoryNewton backward differences

33Lagrange interpolation

35Revision summary 1

38Can You? Checklist 1

41Test exercise 1

42Further problems 1

43

iv Contents

Programme 2

Laplace transforms'!

47

Learning outcomes

47Introduction

48Laplace transforms

48Theorem 1

The first shift theorem

55Theorem 2

Multiplying by t and t"

56Theorem 3

Dividing by t

58Inverse transforms

61Rules of partial fractions

62The 'cover up' rule

66Table of inverse transforms

68Solution of differential equations by Laplace transforms

69Transforms of derivatives

69Solution of first-order differential equations

71Solution of second-order differential equations

74Simultaneous differential equations

81Revision summary 2

87Can You? Checklist 2

89Test exercise 2

90Further problems 2

90

Programme 3 Laplace transforms Z

92

Learning outcomes

92Introduction

93Heaviside unit step function

93Unit step at the origin

94Effect of the unit step function

94Laplace transform of u(t -- c)

97Laplace transform of u(t - c)f(t - c) (the second shift

theorem)

98Revision summary 3

108Can You? Checklist 3

109Test exercise 3

109Further problems 3

110

',Programme 4 Laplace transforms 3Learning outcomes

111Laplace transforms of periodic functions

112Periodic functions

112Inverse transforms

118The Dirac delta function - the unit impulse

122Graphical representation

123Laplace transform of 6(t - a)

124The derivative of the unit step function

127Differential equations involving the unit impulse

128Harmonic oscillators

131

Contents

Damped motion

132Forced harmonic motion with damping

135Resonance

138Revision summary 4

139Can You? Checklist 4

141Test exercise 4

142Further problems 4

143

Programme 5 Ztransforms

144Learning outcomes

144Introduction

145Sequences

145Table of Z transforms

1,48Properties of Z transforms

149Inverse transforms

154Recurrence relations

157Initial terms

158Solving the recurrence relation

159Sampling

163Revision summary 5

166Can You? Checklist 5

168Test exercise 5

169Further problems 5

169

Programme 6

Fourier series

172Learning outcomes

172Introduction

173Periodic functions

173Graphs of y = Asin nx

173Harmonics

174Non-sinusoidal periodic functions

175Analytic description of a periodic function

176Integrals of periodic functions

179Orthogonal functions

183Fourier series

183Dirichlet conditions

186Effects of harmonics

193Gibbs' phenomenon

194Sum of a Fourier series at a point of discontinuity

195Functions with periods other than 27r

197Function with period T

197Fourier coefficients

198Odd and even functions

201Products of odd and even functions

204Half-range series

212Series containing only odd harmonics or only even

harmonics

216

Vi Contents

Significance of the constant term i ao

219Half-range series with arbitrary period

220Revision summary 6

223Can You? Checklist 6

225Test exercise 6

227Further problems 6

223

Programme 7 Introduction to the

231Fourier transform

Learning outcomes

231Complex Fourier series

232introduction

232Complex exponentials

232Complex spectra

237The two domains

238Continuous spectra

239Fourier's integral theorem

247.Some special functions and their transforms

24`1Even functions

241Odd functions

244Top-hat function

246The Dirac delta

248The triangle function

260Alternative forms

261Properties of the Fourier transform

261Linearity

251Time shifting

252Frequency shifting

252Time scaling

263Symmetry

253Differentiation

254The Heaviside unit step function

255Convolution

257The convolution theorem

258Fourier cosine and sine transforms

261Table o¬ transforms

263Revision summary 7

263Can You? Checklist 7

267Test exercise 7

268Further problems 7

268

'~ Programme8 , . Power series solutions of---

271.

ordinary^ differentialm equationsLearning outcomes

271Higher derivatives

272Leibnitz theorem

275Choice of functions of u and v

277

Contents

Power series solutions

278Leibnitz-Maclaurin method

279Frobenius' method

286Solution of differential equations by the method of Frobenius

286Indicial equation

289Bessel's equation

305Bessel functions

307Graphs of Bessel functions Io(x) and jl (x)

311Legendre's equation

31,1Legendre polynomials

311Rodrigue's formula and the generating function

312Sturm-Liouville systems

315Orthogonality

316Legendre's equation revisited

317Polynomials as a finite series of Legendre polynomials

318Revision summary 8

319Can You? Checklist 8

323Test exercise 8

324Further problems 8

324

Programme 9 -..:Numerical solutionso

327,ordinary differential equations

Learning outcomes

327Introduction

328Taylor's series

328Function increment

329First-order differential equations

330Ruler's method

330The exact value and the errors

339Graphical interpretation of Ruler's method

343The Ruler-Cauchy method - or the improved Ruler method

345Ruler-Cauchy calculations

346Runge-Kutta method

351Second-order differential equations

355Ruler second-order method

355Runge-Kutta method for second-order differential equations

357Predictor-corrector methods

362Revision summary 9

365Can You? Checklist 9

367Test exercise 9

367Further problems 9

368

Programme 10

Partial differentiation

-

~_.-

370 3

Learning outcomes

370Small increments

371Taylor's theorem for one independent variable

371Taylor's theorem for two independent variables

371

Ail Contents

Small increments

373Rates of change

375Implicit functions

376Change of variables

377Inverse functions

382General case

384Stationary values of a function

390Maximum and minimum values

391Saddle point

398Lagrange undetermined multipliers

403Functions with two independent variables

403Functions with three independent variables

405Revision summary 10

409Can You? Checklist 10

410Test exercise 10

411Further problems 10

412

Programme 11

Partial differential equations

414Learning outcomes

414Introduction

415Partial differential equations

41.6Solution by direct integration

416Initial conditions and boundary conditions

417The wave equation

418Solution of the wave equation

419Solution by separating the variables

419The heat conduction equation for a uniform finite bar

428Solutions of the heat conduction equation

429Laplace's equation

434Solution of the Laplace equation

435Laplace's equation in plane polar coordinates

439The problem

440Separating the variables

441The n = 0 case

444Revision summary 11

446Can You? Checklist 11

447Test exercise 11

448Further problems 11

449

Programme 12 Matrix algebra

451Learning outcomes

451Singular and non-singular matrices

452Rank of a matrix

453Elementary operations and equivalent matrices

454Consistency of a set of equations

458Uniqueness of solutions

459

Solution of sets of equations

463Inverse method

463Row transformation method

467Gaussian elimination method

471Triangular decomposition method

474Comparison of methods

480Eigenvalues and eigenvectors

480Cayley-Hamilton theorem

487Systems of first-order ordinary differential equations

488Diagonalisation of a matrix

493Systems of second-order differential equations

498Matrix transformation

505Rotation of axes

507Revision summary 12

509Can You? Checklist 12

512Test exercise 12

513Further problems 12

514

Programme 't 3

Numerical solutions of partial

51.7differential equations

Learning outcomes

517Introduction

518Numerical approximation to derivatives

518Functions of two real variables

521Grid values

522Computational molecules

525Summary of procedures

529Derivative boundary conditions

532Second-order partial differential equations

536Second partial derivatives

537Time-dependent equations

542The Crank-Nicolson procedure

547Dimensional analysis

554Revision summary 13

555Can You? Checklist 13

559Test exercise 13

560Further problems 13

561

Programme 14 Multiple integration '!

.566

Learning outcomes

566Introduction

567Differentials

575Exact differential

578Integration of exact differentials

579Area enclosed by a closed curve

581Line integrals

585Alternative form of a line integral

586

x Contents

Properties of line integrals

589Regions enclosed by closed curves

591Line integrals round a closed curve

592Line integral with respect to arc length

596Parametric equations

597Dependence of the line integral on the path of integration

598Exact differentials in three independent variables

603Green's theorem

604Revision summary 14

611Can You? Checklist 14

613Test exercise 14

514Further problems 14

615

Programme I5

Multiple integration 2

617

Learning outcomes

617Double integrals

618Surface integrals

623Space coordinate systems

629Volume integrals

634Change of variables in multiple integrals

643Curvilinear coordinates

645Transformation in three dimensions

653Revision summary 15

655Can You? Checklist 15

657Test exercise 15

6S8Further problems 15

658

Programme 16

Integral functions

661Learning outcomes

661Integral functions

662The gamma function

662The beta function

670Relation between the gamma and beta functions

674Application of gamma and beta functions

676Duplication formula for gamma functions

679The error function

680The graph of erf (x)

681The complementary error function erfc (x)

681Elliptic functions

683Standard forms of elliptic functions

684Complete elliptic functions

684Alternative forms of elliptic functions

688Revision summary 16

691Can You? Checklist 16

698Test exercise 16

694Further problems 16

694

Programme 17 Vector analysis 1 6971Learning outcomes

697Introduction

698Triple products

703Properties of scalar triple products

704Coplanar vectors

705Vector triple products of three vectors

707Differentiation of vectors

710Differentiation of sums and products of vectors

715Unit tangent vectors

71.5Partial differentiation of vectors

718Integration of vector functions

718Scalar and vector fields

721Grad (gradient of a scalar field)

721Directional derivatives

724Unit normal vectors

727Grad of sums and products of scalars

729Div (divergence of a vector function)

731Curl (curl of a vector function)

732Summary of grad, div and curl

733Multiple operations

735Revision summary 17

738Can You? Checklist 17

740Test exercise 17

741Further problems 17

741

Programme IS Vector analysis 2

Learning outcomes

744Line integrals

745Scalar field

745Vector field

748Volume integrals

752Surface integrals

756Scalar fields

757Vector fields

760Conservative vector fields

765Divergence theorem (Gauss' theorem)

770Stokes' theorem

776Direction of unit normal vectors to a surface S

779Green's theorem

785Revision summary 18

788Can You? Checklist 18

790Test exercise 18

791Further problems 18

792

xii Contents

Programme 19 Vector analysis 3

795

Learning outcomes

795Curvilinear coordinates

796Orthogonal curvilinear coordinates

800Orthogonal coordinate systems in space

801Scale factors

805Scale factors for coordinate systems

806General curvilinear coordinate system (u, v, w)

808Transformation equations

809Element of arc ds and element of volume dV in orthogonal

curvilinear coordinates

810Grad, div and curl in orthogonal curvilinear coordinates

811Particular orthogonal systems

81,4Revision summary 19

816Can You? Checklist 19

818Test exercise 19

819Further problems 19

820

Programme 20 Complex analysis 1

821Learning outcomes

821Functions of a complex variable

822Complex mapping

823Mapping of a straight line in the z-plane onto the w-plane

under the transformation w = f(z)

825Types of transformation of the form w = az + b

829Non-linear transformations

838Mapping of regions

843Revision summary 20

857Can You? Checklist 20

858Test exercise 20

858Furtherproblems 20

859

Programme 21

Complex analysis 2

861Learning outcomes

861Differentiation of a complex function

862Regular function

863Cauchy-Riemann equations

865Harmonic functions

867Complex integration

872Contour integration - line integrals in the z-plane

872Cauchy's theorem

875Deformation of contours at singularities

880Conformal transformation (conformal mapping)

889Conditions for conformal transformation

889Critical points

890

Contents

Schwarz-Christoffel transformation

893Open polygons

898Revision summary 21

904Can You? Checklist 21

905Test exercise 21

906Further problems 21

907

Programme 22

Complex analysis 3

909'Learning outcomes

909Maclaurin series

910Radius of convergence

914Singular points

915Poles

915Removable singularities

916Circle of convergence

916Taylor's series

917Laurent's series

919Residues

923Calculating residues

925Integrals of real functions

926Revision summary 22

933Can You? Checklist 22

935Test exercise 22

936Further problems 22

937

', Programme 23

OptimizationnandW linear

9401programming

Learning outcomes

940Optimization

941Linear programming (or linear optimization)

941Linear inequalities

942Graphical representation of linear inequalities

942The simplex method

948Setting up the simplex tableau

948Computation of the simplex

950Simplex with three problem variables

958Artificial variables

962Minimisation

973Applications

977Revision summary 23

981Can You? Checklist 23

982Test exercise 23

983Further problems 23

984Appendix

989Answers

998Index

1027