Engineering Mathematics With Examples And Applications
Engineering Mathematicswith Examplesand ApplicationsXin-She YangMiddlesex UniversitySchool of Science and TechnologyLondon, United Kingdom
Academic Press is an imprint of Elsevier125 London Wall, London EC2Y 5AS, United Kingdom525 B Street, Suite 1800, San Diego, CA 92101-4495, United States50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United StatesThe Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United KingdomCopyright 2017 Elsevier Inc. All rights reserved.No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or anyinformation storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about thePublisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can befound at our website: www.elsevier.com/permissions.This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).NoticesKnowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods,professional practices, or medical treatment may become necessary.Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, orexperiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whomthey have a professional responsibility.To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons orproperty as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in thematerial herein.Library of Congress Cataloging-in-Publication DataA catalog record for this book is available from the Library of CongressBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British LibraryISBN: 978-0-12-809730-4For information on all Academic Press publicationsvisit our website at https://www.elsevier.comPublisher: Nikki LevyAcquisition Editor: Graham NisbetEditorial Project Manager: Susan IkedaProduction Project Manager: Mohanapriyan RajendranDesigner: Matthew LimbertTypeset by VTeX
ContentsAbout the AuthorPrefaceAcknowledgmentixxixiiiPart IFundamentals1. Equations and Functions1.1. Numbers and Real Numbers1.1.1. Notes on Notations andConventions1.1.2. Rounding Numbers and SignificantDigits1.1.3. Concept of Sets1.1.4. Special Sets1.2. Equations1.2.1. Modular Arithmetic1.3. Functions1.3.1. Domain and Range1.3.2. Linear Function1.3.3. Modulus Function1.3.4. Power Function1.4. Quadratic Equations1.5. Simultaneous EquationsExercises3356881011121214151619202. Polynomials and Roots2.1. Index Notation2.2. Floating Point Numbers2.3. Polynomials2.4. RootsExercises21232425283. Binomial Theorem and Expansions3.1. Binomial Expansions3.2. Factorials3.3. Binomial Theorem and Pascal’s TriangleExercises313233354. Sequences4.1. Simple Sequences4.1.1. Arithmetic Sequence4.1.2. Geometric Sequence4.2. Fibonacci Sequence4.3. Sum of a Series4.4. Infinite SeriesExercises373839404144455. Exponentials and Logarithms5.1. Exponential Function5.2. Logarithm5.3. Change of Base for LogarithmExercises474853546. Trigonometry6.1. Angle6.2. Trigonometrical Functions6.2.1. Identities6.2.2. Inverse6.2.3. Trigonometrical Functions of TwoAngles6.3. Sine Rule6.4. Cosine RuleExercises5557585959626263Part IIComplex Numbers7. Complex Numbers7.1.7.2.7.3.7.4.7.5.Why Do Need Complex Numbers?Complex NumbersComplex AlgebraEuler’s FormulaHyperbolic Functions7.5.1. Hyperbolic Sine and Cosine7.5.2. Hyperbolic Identities7.5.3. Inverse Hyperbolic FunctionsExercises676768717272747475v
vi ContentsPart IIIVectors and Matrices14. Multiple Integrals and SpecialIntegrals8. Vectors and Vector Algebra8.1. Vectors8.2. Vector Algebra8.3. Vector Products8.4. Triple Product of VectorsExercises79808385869. Matrices9.1.9.2.9.3.9.4.9.5.MatricesMatrix Addition and MultiplicationTransformation and InverseSystem of Linear EquationsEigenvalues and Eigenvectors9.5.1. Distribution of Eigenvalues9.5.2. Definiteness of a MatrixExercises11111411712011. Integration12112512813012. Ordinary Differential Equations12.1. Differential Equations12.2. First-Order Equations12.3. Second-Order Equations12.4. Higher-Order ODEs12.5. System of Linear ODEsExercises13113213614214314413. Partial Differentiation13.1. Partial Differentiation13.2. Differentiation of Vectors13.3. Polar Coordinates13.4. Three Basic OperatorsExercises15315315515715715815916115.1. Analytic Functions15.2. Complex Integrals15.2.1. Cauchy’s Integral Theorem15.2.2. Residue TheoremExercises163165166168169Part VFourier and Laplace Transforms16. Fourier Series and Transform10. Differentiation11.1. Integration11.2. Integration by Parts11.3. Integration by SubstitutionExercisesLine IntegralMultiple IntegralsJacobianSpecial Integrals14.4.1. Asymptotic Series14.4.2. Gaussian Integrals14.4.3. Error FunctionsExercises15. Complex Integrals8790939899104107108Part IVCalculus10.1. Gradient and Derivative10.2. Differentiation Rules10.3. Series Expansions and Taylor 16.1. Fourier Series16.1.1. Fourier Series16.1.2. Orthogonality16.1.3. Determining the Coefficients16.2. Fourier Transforms16.3. Solving Differential Equations UsingFourier Transforms16.4. Discrete and Fast Fourier TransformsExercises17317317517617918218318517. Laplace Transforms17.1. Laplace Transform17.1.1. Laplace Transform Pairs17.1.2. Scalings and Properties17.1.3. Derivatives and Integrals17.2. Transfer Function17.3. Solving ODE via Laplace Transform17.4. Z-Transform17.5. Relationships between Fourier, Laplaceand rt VIStatistics and Curve Fitting18. Probability and Statistics18.1. Random Variables201
Contents18.2. Mean and Variance18.3. Binomial and Poisson Distributions18.4. Gaussian Distribution18.5. Other Distributions18.6. The Central Limit Theorem18.7. Weibull DistributionExercises20220320720921121221419. Regression and Curve Fitting19.1. Sample Mean and Variance19.2. Method of Least Squares19.2.1. Maximum Likelihood19.2.2. Linear Regression19.3. Correlation Coefficient19.4. Linearization19.5. Generalized Linear Regression19.6. Hypothesis Testing19.6.1. Confidence Interval19.6.2. Student’s t -Distribution19.6.3. Student’s t Part VIINumerical Methods20. Numerical Methods20.1.20.2.20.3.20.4.20.5.Finding RootsBisection MethodNewton-Raphson MethodNumerical IntegrationNumerical Solutions of ODEs20.5.1. Euler Scheme20.5.2. Runge-Kutta MethodExercises23123223323423723723724121. Computational Linear Algebra21.1.21.2.21.3.21.4.System of Linear EquationsGauss EliminationLU FactorizationIteration Methods21.4.1. Jacobi Iteration Method21.4.2. Gauss-Seidel Iteration21.4.3. Relaxation Method21.5. Newton-Raphson Method21.6. Conjugate Gradient rt VIIIOptimization22. Linear Programming22.1. Linear Programming22.2. Simplex Method22.2.1. Basic Procedure22.2.2. Augmented Form22.3. A Worked ExampleExercises25926026126226326523. Optimization23.1. Optimization23.2. Optimality Criteria23.2.1. Feasible Solution23.2.2. Optimality Criteria23.3. Unconstrained Optimization23.3.1. Univariate Functions23.3.2. Multivariate Functions23.4. Gradient-Based Methods23.4.1. Newton’s Method23.4.2. Steepest Descent Method23.5. Nonlinear Optimization23.5.1. Penalty Method23.5.2. Lagrange Multipliers23.6. Karush-Kuhn-Tucker Conditions23.7. Sequential Quadratic Programming23.7.1. Quadratic Programming23.7.2. Sequential 275275276278278279280281281282283Part IXAdvanced Topics24. Partial Differential er PDEsClassification of Second-Order PDEsClassic Mathematical Models: SomeExamples24.4.1. Laplace’s and Poisson’sEquation24.4.2. Parabolic Equation24.4.3. Hyperbolic Equation24.5. Solution Techniques24.5.1. Separation of Variables24.5.2. Laplace Transform24.5.3. Fourier Transform24.5.4. Similarity Solution24.5.5. Change of 6297298299
viiiContents25. Tensors25.1. Summation Notations25.2. Tensors25.2.1. Rank of a Tensor25.2.2. Contraction25.2.3. Symmetric and AntisymmetricTensors25.2.4. Tensor Differentiation25.3. Hooke’s Law and 1131431631731931932032132132232428. Mathematical Modeling28.1.28.2.28.3.28.4.28.5.Mathematical ModelingModel FormulationDifferent Levels of ApproximationsParameter EstimationTypes of Mathematical Models28.5.1. Algebraic Equations28.5.2. Tensor Relationships28.5.3. Differential Equations: ODE on and IntegrationComplex NumbersVectors and MatricesFourier Series and TransformAsymptoticsSpecial Integrals341341341342343343B. Mathematical Software Packages27. Integral Equations27.1. Integral Equations27.1.1. Fredholm Integral Equations27.1.2. Volterra Integral Equation27.2. Solution of Integral Equations27.2.1. Separable Kernels27.2.2. Volterra EquationExercises335336337337338A. Mathematical Formulas26. Calculus of Variations26.1. Euler-Lagrange Equation26.1.1. Curvature26.1.2. Euler-Lagrange Equation26.2. Variations with Constraints26.3. Variations for Multiple VariablesExercises28.5.4. Functional and IntegralEquations28.5.5. Statistical Models28.5.6. Fuzzy Models28.5.7. Learned Models28.5.8. Data-Driven Models28.6. Brownian Motion and Diffusion:A Worked ExampleExercises325326328330332332333333B.1. MatlabB.1.1. MatlabB.1.2. MuPADB.2. Software Packages Similar to MatlabB.2.1. OctaveB.2.2. ScilabB.3. Symbolic Computation PackagesB.3.1. MathematicaB.3.2. MapleB.3.3. MaximaB.4. R and PythonB.4.1. RB.4.2. Python345345346347347348348348348349350350350C. Answers to ExercisesBibliographyIndex381383
Chapter 28Mathematical ModelingChapter Points Mathematical modeling is introduced with the basic modeling procedure, including mathematical model formulation based onphysical laws, parameter estimation and normalization. Different levels of approximations are explained to discuss the assumptions, abstractions and the balance of accuracy and modelcomplexity. Different types of models are explained with some examples relevant to science and engineering applications. A worked example is presented in detail to model Brownian motion and diffusion.28.1 MATHEMATICAL MODELINGMathematical modeling is the process of formulating an abstract model in terms of mathematical language to describethe complex behavior of a real system. Mathematical models are quantitative models and often expressed in terms ofordinary differential equations and partial differential equations. Mathematical models can also be statistical models, fuzzylogic models and empirical relationships. In fact, any model description using mathematical language can be called amathematical model. Mathematical modeling is widely used in natural sciences, computing, engineering, meteorology,economics and finance. For example, theoretical physics is essentially all about the modeling of real-world processes usingseveral basic principles (such as the conservation of energy and momentum) and a dozen important equations (such as thewave equation, the Schrödinger equation, the Einstein equation). Most of these equations are partial differential equations.An important feature of mathematical modeling is its interdisciplinary nature. It involves applied mathematics, computersciences, physics, chemistry, engineering, biology and other disciplines such as economics, depending on the problem ofinterest. Mathematical modeling in combination with scientific computing is an emerging interdisciplinary technology.Many international companies use it to model physical processes, to design new products, to find solutions to challengingproblems, and to increase their competitiveness in international markets.Example 28.1One of the simplest models we learned in school is probably Newton’s second law that relates the force F acted on a body with amass m to its acceleration a. That isF ma,which is one of the most accurate models in science. This is a linear relationship and thus a linear model, but a very well-testedmodel.Apart from a simple mathematical formula, as a mathematical model, all the quantities involved such as force, mass andacceleration must have appropriate units. For example, the unit of F is Newton (N), the unit of mass is kilogram (kg), while theacceleration has a derived unit (a combination of units) of m/s2 . Therefore, a person of 80 kg has a weight (the force acted upon theperson by the Earth) is W mg where g 9.8 m/s2 is the acceleration due to gravity. That isW mg 80 (kg) 9.8 (m/s2 ) 784 N.If the units are wrong, even a good model will give wrong values. This highlights the importance of units and the parameters (e.g.,g here) in mathematical modeling.Engineering Mathematics with Examples and ApplicationsCopyright 2017 Elsevier Inc. All rights reserved.325
326 PART IX Advanced TopicsFIGURE 28.1 Mathematical modeling.Mathematical modeling is an iterative, multidisciplinary process with many steps from the abstraction of the processesin nature to the construction of the full mathematical models. The basic steps of mathematical modeling can be summarizedas meta-steps shown in Fig. 28.1. The process typically starts with the analysis of a real world problem so as to extract thefundamental physical processes by idealization and various assumptions. Once an idealized physical model is formulated, itcan then be translated into the corresponding mathematical model in terms of partial differential equations (PDEs), integralequations, and statistical models. Then, the mathematical model should be investigated in great detail by mathematicalanalysis (if possible), numerical simulations and other tools so as to make predictions under appropriate conditions. Then,these simulation results and predictions will be validated against the existing models, well-established benchmarks, andexperimental data. If the results are satisfactory (but they rarely are at first), then the mathematical model can be accepted.If not, both the physical model and mathematical model will be modified based on the feedback, then the new simulationsand prediction will be validated again.After a certain number of iterations of the whole process (often many), a good mathematical model can properly beformulated, which will provide great insight into the real world problem and may also predict the behavior of the processunder study.For any physical problem in physics and engineering, for example, there are traditionally two ways to deal with it byeither theoretical approaches or field observations and experiments.The theoretical approach in terms of mathematical modeling is an idealization and simplification of the real problemand the theoretical models often extract the essential or major characteristics of the problem. The mathematical equationsobtained even for such over-simplified systems are usually very difficult for mathematical analysis. On the other hand, thefield studies and experimental approach can be expensive if not impractical. Apart from financial and practical limitations,other constraining factors include the inaccessibility of the locations, the range of physical parameters, and time for carryingout various experiments. As the computing speed and power of computers have increased dramatically in the last fewdecades, a practical third way or approach is emerging, which is computational modeling and numerical experimentationbased on mathematical models. It is now widely acknowledged that computational modeling and computer simulationsserve as a cost-effective alternative, bridging the gap between theory and practice as well as complementing the traditionaltheoretical and experimental approaches to problem solving.Mathematical modeling is essentially an abstract art of formulating the mathematical models from their correspondingreal-world problems. The mastery of this art requires practice and experience, and it is not easy to teach such skills as thestyle of mathematical modeling largely depends on each person’s own insight, abstraction, type of problems, and experienceof dealing with similar problems. Even for the same physical process, different models could be obtained, depending onthe emphasis of some part of the process, say, based on your interest in certain quantities in a particular problem, while thesame quantities could be viewed as unimportant in other processes and other problems.28.2 MODEL FORMULATIONMathematical modeling often starts with the analysis of the physical process and attempts to make an abstract physicalmodel by idealization and approximations. From this idealized physical model, we can use the various first principles suchas the conservation of mass, momentum, energy and Newton’s laws to translate into mathematical equations. However,such transformation from practice to theory can rarely be achieved in a single step, thus an iterative loop between theoryand practice is needed, as pointed out by the famous statistician George Box.As an example, let us now look at the example of the diffusion process of sugar in a glass of water. We know that thediffusion of sugar will occur if there is any spatial difference in the sugar concentration. The physical process is complicated
Mathematical Modeling Chapter 28327FIGURE 28.2 Representative element volume (REV).and many factors could affect the distribution of sugar concentration in water, including the temperature, stirring, mass ofsugar, type of sugar, how you add the sugar, even geometry of the container and others. We can idealize the process byassuming that the temperature is constant (so as to neglect the effect of heat transfer), and that there is no stirring becausestirring will affect the effective diffusion coefficient and introduce the advection of water or even vertices in the (turbulent)water flow.We then choose a representative element volume (REV) whose size is very small compared with the size of the cup sothat we can use a single value of concentration to represent the sugar content inside this REV. If this REV is too large, thereis a considerable variation in sugar concentration inside this REV. We also assume that there is no chemical reaction betweensugar and water (otherwise, we are dealing with something else). If you drop the sugar into the cup from a considerableheight, the water inside the glass will splash and thus fluid volume will change, and this becomes a fluid dynamics problem.So we are only interested in the process after the sugar is added and we are not interested in the initial impurity of the water(to a certain degree).With these assumptions, the whole process is now idealized as the physical model of the diffusion of sugar in still waterat a constant temperature. Now we have to translate this idealized model into a mathematical model, and in the presentcase, a parabolic partial differential equation or diffusion equation.Let c be the averaged concentration in a representative element volume with a volume dV inside the cup, and let be an arbitrary, imaginary closed volume (much larger than our REV but smaller than the container, see Fig. 28.2). Weknow that the rate of change of the mass of sugar per unit time inside is%%% δ1 cdV ,(28.1) t where t is time. As the mass is conserved, this change of sugar content in
Engineering Mathematics with Examples and Applications Xin-She Yang Middlesex University School of Science and Technology London, United Kingdom. Academic Press is an imprint of Elsevier
IBDP MATHEMATICS: ANALYSIS AND APPROACHES SYLLABUS SL 1.1 11 General SL 1.2 11 Mathematics SL 1.3 11 Mathematics SL 1.4 11 General 11 Mathematics 12 General SL 1.5 11 Mathematics SL 1.6 11 Mathematic12 Specialist SL 1.7 11 Mathematic* Not change of base SL 1.8 11 Mathematics SL 1.9 11 Mathematics AHL 1.10 11 Mathematic* only partially AHL 1.11 Not covered AHL 1.12 11 Mathematics AHL 1.13 12 .
as HSC Year courses: (in increasing order of difficulty) Mathematics General 1 (CEC), Mathematics General 2, Mathematics (‘2 Unit’), Mathematics Extension 1, and Mathematics Extension 2. Students of the two Mathematics General pathways study the preliminary course, Preliminary Mathematics General, followed by either the HSC Mathematics .
2. 3-4 Philosophy of Mathematics 1. Ontology of mathematics 2. Epistemology of mathematics 3. Axiology of mathematics 3. 5-6 The Foundation of Mathematics 1. Ontological foundation of mathematics 2. Epistemological foundation of mathematics 4. 7-8 Ideology of Mathematics Education 1. Industrial Trainer 2. Technological Pragmatics 3.
Materials Science and Engineering, Mechanical Engineering, Production Engineering, Chemical Engineering, Textile Engineering, Nuclear Engineering, Electrical Engineering, Civil Engineering, other related Engineering discipline Energy Resources Engineering (ERE) The students’ academic background should be: Mechanical Power Engineering, Energy .
Advanced Engineering Mathematics Dr. Elisabeth Brown c 2019 1. Mathematics 2of37 Fundamentals of Engineering (FE) Other Disciplines Computer-Based Test (CBT) Exam Specifications. Mathematics 3of37 1. What is the value of x in the equation given by log 3 2x 4 log 3 x2 1? (a) 10 (b) 1(c)3(d)5 E. Brown . Mathematics 4of37 2. Consider the sets X and Y given by X {5, 7,9} and Y { ,} and the .
3.1 EGR 101: Introductory Mathematics for Engineering Applications The WSU model begins with the development of EGR 101, "Introductory Mathematics for Engineering Applications," a novel freshman-level engineering mathematics course. The goal of EGR 101 is to address only the salient mathematics topics actually used in the primary core
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1) Grewal B.S. - Higher Engineering Mathematics, 40/e, Khanna Publishers. 2) Kreyszig E.K. - Advanced Engineering Mathematics, John Wiley. 3) Ramana B.V. - Higher Engineering Mathematics, (TMH) 4) Singh R.R. & Bhatt M. - Engineering Mathematics, (TMH) I A 2 ENGINEERING PHYSICS Aim : To enabl
