Engineering Mathematics: Programs and Problems PDF
Preview Engineering Mathematics: Programs and Problems
ENGINEERING 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
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