Michael Batty Essential Engineering Mathematics Download free ebooks at bookboon.com Essential Engineering Mathematics © 2011 Michael Batty & Ventus Publishing ApS ISBN 978-87-7681-735-0 Download free ebooks at bookboon.com Contents Essential Engineering Mathematics Contents Introduction 1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.3.1 1.3.2 1.3.3 1.3.4 Preliminaries Number Systems: The Integers, Rationals and Reals Working with the Real Numbers Intervals Solving Inequalities Absolute Value Inequalities Involving Absolute Value Complex Numbers Imaginary Numbers The Complex Number System and its Arithmetic Solving Polynomial Equations Using Complex Numbers Geometry of Complex Numbers 10 10 13 13 13 14 16 17 17 18 19 20 2.1 2.2 2.2.1 2.2.2 2.2.3 2.3 Vectors and Matrices Vectors Matrices and Determinants Arithmetic of Matrices Inverse Matrices and Determinants The Cross Product Systems of Linear Equations and Row Reduction 23 23 26 26 27 28 30 Please click the advert The next step for 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of Linear Equations Row Reduction Finding the Inverse of a Matrix using Row Reduction Bases Eigenvalues and Eigenvectors 30 30 32 34 35 3.1 3.1.1 3.1.2 3.1.3 3.2 3.3 Functions and Limits Functions Denition of a Function Piping Functions Together Inverse Functions Limits Continuity 39 39 39 41 42 43 48 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Calculus of One Variable Part 1: Differentiation Derivatives The Chain Rule Some Standard Derivatives Dierentiating Inverse Functions Implicit Differentiation Logarithmic Differentiation Higher Derivatives L’Hôpital’s Rule Taylor Series 51 51 56 58 59 61 63 66 67 69 Please click the advert Join EADS A global leader in aerospace, defence and related services Let your imagination take shape EADS unites a leading aircraft manufacturer, the world’s largest helicopter supplier, a global leader in space programmes and a worldwide leader in global security solutions and systems to form Europe’s largest defence and aerospace group More than 140,000 people work at Airbus, Astrium, Cassidian and Eurocopter, in 90 locations globally, to deliver some of the industry’s most exciting projects learning and development opportunities, and all the support you need, you will tackle interesting challenges on state-of-the-art products We take more than 5,000 interns every year across disciplines ranging from engineering, IT, procurement and finance, to strategy, customer support, marketing and sales Positions are available in France, Germany, Spain and the UK An EADS internship offers the chance to use your theoretical knowledge and apply it first-hand to real situations and assignments during your studies Given a high level of responsibility, plenty of To find out more and apply, visit www.jobs.eads.com You can also find out more on our EADS Careers Facebook page Download free ebooks at bookboon.com Contents Essential Engineering Mathematics 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Calculus of One Variable Part 2: Integration Summing Series Integrals Antiderivatives Integration by Substitution Partial Fractions Integration by Parts Reduction Formulae Improper Integrals 79 79 81 85 88 93 98 101 105 6.1 6.2 6.3 6.4 6.5 6.6 Calculus of Many Variables Surfaces and Partial Derivatives Scalar Fields Vector Fields Jacobians and the Chain Rule Line Integrals Surface and Volume Integrals 111 111 113 115 117 119 121 7.1 7.2 Ordinary Differential Equations First Order Dierential Equations Solvable by Integrating Factor First Order Separable Differential Equations 123 123 126 7.3 Second Order Linear Differential Equations with Constant Coeficients: The Homogeneous Case 126 Excellent Economics and Business programmes at: Please click the advert “The perfect start of a successful, international career.” CLICK HERE to discover why both socially and academically the University of Groningen is one of the best places for a student to be www.rug.nl/feb/education Download free ebooks at bookboon.com Contents Essential Engineering Mathematics 7.4 7.5 Second Order Linear Differential Equations with Constant Coeficients: The Inhomogeneous Case Initial Value Problems 131 132 8.1 8.2 8.3 Complex Function Theory Standard Complex Functions The Cauchy-Riemann Equations Complex Integrals 134 135 140 142 Index 147 Please click the advert Teach with the Best Learn with the Best Agilent offers a wide variety of affordable, industry-leading electronic test equipment as well as knowledge-rich, on-line resources —for professors and students We have 100’s of comprehensive web-based teaching tools, lab experiments, application notes, brochures, DVDs/ CDs, posters, and more See what Agilent can for you www.agilent.com/find/EDUstudents www.agilent.com/find/EDUeducators © Agilent Technologies, Inc 2012 u.s 1-800-829-4444 canada: 1-877-894-4414 Download free ebooks at bookboon.com Introduction Essential Engineering Mathematics Introduction This book is partly based on lectures I gave at NUI Galway and Trinity College Dublin between 1998 and 2000 It is by no means a comprehensive guide to all the mathematics an engineer might encounter during the course of his or her degree The aim is more to highlight and explain some areas commonly found difficult, such as calculus, and to ease the transition from school level to university level mathematics, where sometimes the subject matter is similar, but the emphasis is usually different The early sections on functions and single variable calculus are in this spirit The later sections on multivariate calculus, differential equations and complex functions are more typically found on a first or second year undergraduate course, depending upon the university The necessary linear algebra for multivariate calculus is also outlined More advanced topics which have been omitted, but which you will certainly come across, are partial differential equations, Fourier transforms and Laplace transforms Download free ebooks at bookboon.com Introduction Essential Engineering Mathematics This short text aims to be somewhere first to look to refresh your algebraic techniques and remind you of some of the principles behind them I have had to omit many topics and it is unlikely that it will cover everything in your course I have tried to make it as clean and uncomplicated as possible Hopefully there are not too many mistakes in it, but if you find any, have suggestions to improve the book or feel that I have not covered something which should be included please send an email to me at batty.mathmo@googlemail.com Michael Batty, Durham, 2010 Download free ebooks at bookboon.com Essential Engineering Mathematics Chapter Preliminaries 1.1 Number Systems: The Integers, Rationals and Reals Calculus is a part of the mathematics of the real numbers You will probably be used to the idea of real numbers, as numbers on a “line” and working with graphs of real functions in the product of two lines, i.e a plane To define rigorously what real numbers are is not a trivial matter Here we will mention two important properties: • • Download free ebooks at bookboon.com 10 ...Michael Batty Essential Engineering Mathematics Download free ebooks at bookboon.com Essential Engineering Mathematics © 2011 Michael Batty & Ventus Publishing... free ebooks at bookboon.com Essential Engineering Mathematics Chapter Preliminaries 1.1 Number Systems: The Integers, Rationals and Reals Calculus is a part of the mathematics of the real numbers... Masters in Management 2010 employment report Download free ebooks at bookboon.com Contents Essential Engineering Mathematics 2.3.1 2.3.2 2.3.3 2.4 2.5 Systems of Linear Equations Row Reduction Finding