Table Of ContentENGINEERING MATHEMATICS
Companion volume
K. A. Stroud Further Engineering Mathematics. 2nd edition
Engineering
Mathematics
Programs and Problems
Ken A. Stroud
formerly Principal Lecturer in
Mathematics
Lanchester Polytechnic
Coventry, England
Third Edition
Springer-Verlag
© K. A. Stroud 1970, 1982, 1987
All right£ reserved. No part of this publication
may be reproduced or transmitted, in any form or
by any means, without permission.
First edition 1970
Second edition 1982
Third edition 1991
Published in the UK by
The Macmillan Press Ltd,
London and Basingstoke
Sole distributors in the USA
Springer-Verlag New York Inc.
175 Fifth Avenue,
New York, NY 10010
USA
ISBN-13: 978-1-4615-9655-4 e-ISBN-13: 978-1-4615-9653-0
DOl: 10.1007/978-1-4615-9653-0
CONTENTS
Preface to the first edition xi
Preface to the second edition xiii
Preface to the third edition xiv
Hints on using the book xv
Useful background information xvi
Programme 1: Complex Numbers, Part 1
Introduction: The symbol j; powers ofj ; complex numbers
Multiplication of complex numbers
Equal complex numbers
Graphical representation of a complex number
Graphical addition of complex numbers
Polar form of a complex number
Exponential form of a complex number
Test exercise I
Further problems I
Programme 2: Complex Numbers, Part 2 37
Introduction
Loci problems
Test exercise II
Further problems II
Programme 3: Hyperbolic Functions 73
Introduction
Graph!; of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Trig. identities and hyperbolic identities
Relationship between trigonometric & hyperbolic functions
Test exercise III
Further problems III
Programme 4: Determinants 101
Determinants
Determinants of the third order
Evaluation of a third order determinant
Simultaneous equations in three unknowns
Consistency of a set ofe quations
v
Properties of determinants
Test exercise IV
Further problems IV
Programme 5: Matrices 141
Definitions; order; types of matrices
Operations
Transpose and inverse of a square matrix
Solution of sets of linear equations
Gaussian elimination method
Eigenvalues and eigenvectors
Revision summary
Test exercise V
Further problems V
Programme 6: Vectors 189
Introduction: Scalar and vector quantities
Vector representation
Two equal vectors
Types of vectors
Addition of vectors
Components of a given vector
Components of a vector in terms of unit vectors
Vectors in space
Direction cosines
Scalar product of two vectors
Vector product of two vectors
Angle between two vectors
Direction ratios
Summary
Test exercise VI
Further problems VI
219
Programme 7: Differentiation
Standard differential coefficients
Functions ofa function
Logarithmic differentiation
Implicit functions
Parametric equations
Test exercise VII
Further problems VII
Programme 8: Differentiation Applications, Part 1 243
Equation of a straight line
vi
Centre of curvature
Test exercise VIII
Further problems VIII
271
Programme 9: Differentiation Applications, Part 2
Inverse trigonometrical functions
Differentiation of inverse trig. functions
Differential coefficients of inverse hyperbolic functions
Maximum and minimum values (turning points)
Test exercise IX
Further problems IX
Programme 10: Partial Differentiation, Part 1 299
Partial differentiation
Small increments
Test exercise X
Further problems X
Programme 11: Partial Differentiation, Part 2 325
Partial differentiation
Rates of change problems
Change of variables
Test exercise XI
Further problems XI
Programme 12: Curves and Curve Fitting 345
Standard curves
Asymptotes
Systematic curve sketching
Curve fitting
Method of least squares
Test exercise XII
Further problems XII
Programme 13: Series, Part 1 395
Sequences and series
Arithmetic and geometric means
Series of powers of natural numbers
Infinite series: limiting values
Convergent and divergent series
Tests for convergence; absolute convergence
Test exercise XIII
Further problems XIII
vii
Programme 14: Series, Part 2 425
s
Power series, Maclaurin series
Standard series
The binomial series
Approximate values
Limiting values
Test exercise XIV
Further problems XIV
455
Programme 15: Integration, Part 1
Introduction
Standard integrals
Functions ofa linear function
Integrals of the form ff(x).['(x)dx etc.
Integration of products - integration by parts
Integration by partial fractions
Integration of trigonometrical functions
Test exercise XV
Further problems XV
Programme 16: Integration, Part 2 487
Test exercise XVI
Further problems XVI
Programme 17: Reduction Formulae 517
Test exercise XVII
Further problems XVII
Programme 18: Integration Applications, Part 1 533
Parametric equations
Mean values
R.m.s. values
Summary sheet
Test exercise XVIII
Further problems XVIII
555
Programme 19: Integration Applications, Part 2
Introduction
Volumes of solids of revolution
Centroid of a plane figure
Centre ofg ravity of a solid of revolution
Lengths of curves
Lengths of curves - parametric equations
Surfaces of revolution
viii
Surfaces of revolution - parametric equations
Rules of Pappus
Revision summary
Test exercise XIX
Further problems XIX
Programme 20: Integration Applications, Part 3 581
Moments of inertia
Radius of gyration
Parallel axes theorem
Perpendicular axes theorem
Useful standard results
Second moment of area
Composite figures
Centres of pressure
Depth of centre of pressure
Test exercise XX
Further problems XX
Programme 21: Approximate Integration 615
Introduction
ApprOXimate integration
Method 1 - by series
Method 2 - Simpson's rule
Proof of Simpson's rule
Test exercise XXI
Further problems XXI
Programme 22: Polar Co-ordinates System 637
Introduction to polar co-ordinates
Polar curves
Standard polar curves
Test exercise XXII
Further problems XXII
Programme 23: Multiple Integrals 663
Summation in two directions
Double integrals: triple integrals
Applications
Alternative notation
Determination of volumes by mUltiple integrals
Test exercise XXIII
Further problems XXIII
Programme 24: First Order Differential Equations 691
Introduction
Formation of differential equations
ix
Solution of differential equations
Method 1 - by direct integration
Method 2 - by separaling the variables
Method 3 - homogeneous equations: by substituting y = vx
Method 4 - linear equations: use of integrating factor
Test exercise XXIV
Further problems XXIV
Programme 25: Second Order Differential Equations with Constant 735
Coefficients
Test exercise XXV
Further problems XXV
Programme 26: Operator D Methods 767
The operator D
Inverse operator I/O
Solution of differential equations by operator 0 methods
Special cases
Test exercise XXVI
Further problems XXVI
Programme 27: Statistics 805
Discrete and continuous data
Grouped data; class boundaries and class interval
Frequency and relative frequency; histograms
Central tendency - mean, mode and median
Coding
Dispersion - range, variance and standard deviation
Frequency polygons and frequency curves
Normal distribution curve; standardised normal curve
Test exercise XXVII
Further problems XXVII
Programme 28: Probability 846
Empirical and classical probability
Addition and multiplication laws of probability
Discrete and continuous probability distributions
Mean and standard deviation of a distribution
Binomial and Poisson distributions
Nonnal distribution curve, standard nonnal curve, areas
under the standard normal curve
Test exercise XXVIII
Further problems XXVIII
Answers 900
Index 946
x
