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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
palgfavemacmittan
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