Calculus for Engineering Students: Fundamentals, Real Problems, and Computers (Mathematics in Science and Engineering)

Calculus for Engineering Students: Fundamentals, Real Problems, and Computers insists that mathematics cannot be separated from chemistry, mechanics, electricity, electronics, automation, and other disciplines. It emphasizes interdisciplinary problems as a way to show the importance of calculus in engineering tasks and problems. While concentrating on actual problems instead of theory, the book uses Computer Algebra Systems (CAS) to help students incorporate lessons into their own studies. Assuming a working familiarity with calculus concepts, the book provides a hands-on opportunity for students to increase their calculus and mathematics skills while also learning about engineering applications. Organized around project-based rather than traditional homework-based learning Reviews basic mathematics and theory while also introducing applications Employs uniform chapter sections that encourage the comparison and contrast of different areas of engineering Front-Matter_2020_Calculus-for-Engineering-Students Copyright_2020_Calculus-for-Engineering-Students Contents_2020_Calculus-for-Engineering-Students Contents List-of-contributors_2020_Calculus-for-Engineering-Students List of contributors About-the-editors_2020_Calculus-for-Engineering-Students About the editors Preface_2020_Calculus-for-Engineering-Students Preface 1---Limits-and-apparent-paradoxes-in-economics_2020_Calculus-for-Engineering 1 Limits and apparent paradoxes in economics and engineering 1.1 Limit of a function 1.1.1 Introduction to limits 1.1.2 Solving strategies 1.1.2.1 Direct substitution 1.1.2.2 Equivalent infinitesimals 1.1.2.3 Indeterminate forms 1.2 Areas of application 1.3 Challenging problems 1.3.1 Compound interest 1.3.2 Computing the perimeter and area of a snowflake 1.3.3 Additional examples of applicability 1.4 Conclusions References 2---Derivative--tool-for-approximation-and-in_2020_Calculus-for-Engineering- 2 Derivative: tool for approximation and investigation 2.1 Derivative: overview of theory 2.2 Derivative in applications 2.3 Exploring derivative 2.3.1 Related rates 2.3.2 Approximating derivative of a function 2.3.3 Approximating functions 2.3.4 Approximating formulas 2.3.5 Investigating solutions of differential equations Answers to exercises References 3---Complex-numbers-and-some-applicatio_2020_Calculus-for-Engineering-Studen 3 Complex numbers and some applications 3.1 Introduction 3.1.1 Complex arithmetic 3.1.2 Properties of complex numbers 3.1.3 Geometric interpretation 3.1.4 The complex logarithm 3.1.5 Important theorems De Moivre's theorem Fundamental theorem of algebra Linear factor theorem Conjugate roots theorem 3.1.6 Roots of complex numbers 3.1.7 Matrix representation 3.2 Illustrations 3.3 Quaternions References 4---Sequences-and-series--a-tool-for-approx_2020_Calculus-for-Engineering-St 4 Sequences and series: a tool for approximation 4.1 Sequences and series: overview of theory 4.2 Sequences and series in applications 4.3 Exploring sequences and series 4.3.1 Asymptotic growth at infinity 4.3.2 Decimal expression of a number 4.3.3 Expressing numbers as sequences and series 4.3.4 Iterative approximation Answers to exercises References 5---Vibrations-and-harmonic-analysis_2020_Calculus-for-Engineering-Students 5 Vibrations and harmonic analysis 5.1 Basic theory background 5.1.1 Taylor series 5.1.2 Fourier series 5.1.2.1 Real forms of Fourier series 5.1.2.2 Complex form of Fourier series 5.2 Fourier series in applications 5.3 Mechanical vibration forced by periodic force with viscous damping: harmonic analysis of a force, stabilized output movement obtained by principle of superposition Problem (a) The system is forced by harmonic function F(t). a1. Solution in exponential form. a2. Solution in real goniometric form. (b) The system is forced by a harmonic function with working hold-ups. Solution in exponential form. Time domain Frequency domain (c) The system is forced by a polygonal periodic chain F(t). Solution in real goniometric form. References 6---Applications-of-integral-calculus_2020_Calculus-for-Engineering-Students 6 Applications of integral calculus 6.1 Key ideas on the calculus of primitive integrals 6.1.1 Methods of integration 6.1.2 The construction of the Riemann integral 6.1.2.1 Operations with integrable functions 6.2 Description of general problems and areas where they are very common 6.2.1 Introductory problems 6.3 Challenging problems 6.3.1 Approximate the mass/position of the gravity center of a bar 6.3.2 Determine the moment of inertia of a wire 6.3.3 Rolling motion in mechanics, remarkable curve 6.3.4 Speed prediction models 6.3.4.1 The problem of estimating the distance traveled References 7---Multiple-integrals-in-mechanical-engin_2020_Calculus-for-Engineering-Stu 7 Multiple integrals in mechanical engineering 7.1 Background Double integrals Double integral calculus Polar coordinates and their relation with Cartesian coordinates Cylindrical coordinates Spherical coordinates Double integrals in polar coordinates Triple integrals Triple integrals calculus Triple integrals in cylindrical and spherical coordinates 7.2 Applications of multiple integrals 7.3 Real problems 7.3.1 Mass center of an object 7.3.2 Viscometer design References 8---Critical-forces-and-collisions--How-to-solve-no_2020_Calculus-for-Engine 8 Critical forces and collisions. How to solve nonlinear equations and their systems 8.1 Preliminaries 8.1.1 General principles for iterative methods - or how to jump from one equation to the system of equations 8.1.2 Newton's method for systems of nonlinear equations again and properly 8.1.3 Convergence of Newton's method. Advantages and disadvantages of Newton's method 8.1.4 Modifications of Newton's algorithm. Globalization using step lengths 8.1.4.1 Properties of the modified Newton's method 8.2 Why the nonlinear systems of equations are important. How important it is to study nonlinear systems 8.3 Nonlinear equations and their systems in applications 8.3.1 Problems of nonlinear equations 8.3.2 Problems on systems of nonlinear equations 8.3.2.1 A simple numerical example of Newton's method for systems of nonlinear equations. Collision avoidance for unmanned vehicles with fixed trajectories 8.3.2.2 Robot arm example 8.3.3 The two-bar truss problem/buckling problem for two bars References 9---Shortest-path-problem-and-computer-alg_2020_Calculus-for-Engineering-Stu 9 Shortest path problem and computer algorithms 9.1 Background 9.1.1 Notation, definitions, and properties 9.1.2 Data structure and labeling correcting algorithms 9.2 Description of general path problems and areas where they are very common 9.3 Real problems 9.3.1 Problem 1: traveling in Portugal 9.3.2 Problem 2: maximum capacity path 9.4 Combined network problems 9.4.1 Problem 3: shortest (minimize cost or time) path on the set of maximum capacity paths References 10---Random-variables-as-arc-parameters-when-solv_2020_Calculus-for-Engineer 10 Random variables as arc parameters when solving shortest path problems 10.1 Background Probabilistic networks Notation, definitions, and properties 10.2 Description of general problems and areas where they are very common 10.2.1 Linear utility function 10.2.2 Quadratic utility function 10.2.3 Exponential utility function 10.3 Real problems References 11---Snails--snakes--and-first-order-ordinary-di_2020_Calculus-for-Engineeri 11 Snails, snakes, and first-order ordinary differential equations 11.1 Background General ideas about Runge-Kutta methods Runge-Kutta methods convergency Linear autonomous systems' stability n-Equation autonomous linear systems stability 11.2 Description of general problems and areas where they are very common A) Geometric applications B) Population growth C) Trajectories of falling bodies and other movement problems D) Radioactive decay E) Newton's law of cooling F) Epidemiology G) Chemistry H) Physics. Serial circuits I) Financial mathematics 11.3 Real problems 11.3.1 The epidemiological, and also malware propagation, model of Kermack and McKendrick Solving the problem 11.3.2 Coevolution and chirality: a story of snails and snakes References 12---Oscillations-in-higher-order-differential-equat_2020_Calculus-for-Engin 12 Oscillations in higher-order differential equations and systems of differential equations 12.1 Basic theory background 12.1.1 Particular types of differential equations of order n>=2 12.1.1.1 Linear equations (a) Homogeneous linear equations (b) Nonhomogeneous linear equations (c) Linear differential equations with constant coefficients 12.1.1.2 Euler equations 12.1.1.3 Nonlinear equations 12.1.2 Systems of differential equations 12.1.2.1 Numerical solution 12.2 Higher-order differential equations in practice 12.3 Challenging problems in applications References 13---Partial-differential-equations_2020_Calculus-for-Engineering-Students 13 Partial differential equations 13.1 Introduction 13.1.1 Some properties of PDEs 13.1.2 First-order PDEs 13.2 Applications of partial differential equations 13.2.1 Second-order PDEs 13.2.2 Wave equation 13.2.3 Heat equation 13.2.4 Laplace's equation 13.2.5 Laplace transforms 13.2.6 Heat equation revisited 13.3 Real engineering problems 14---Laplace-transforms--Engineering-application_2020_Calculus-for-Engineeri 14 Laplace transforms 14.1 Introduction to Laplace transforms 14.1.1 Standard transforms 14.1.2 Inverse Laplace transforms 14.1.3 Partial fractions 14.1.4 The "cover up" rule 14.1.5 The first shift theorem 14.2 Solving first- and second-order differential equations 14.2.1 Transforms of derivatives 14.2.2 Alternative notation 14.2.3 Solving first-order differential equations 14.2.4 Solving second-order differential equations using Laplace transforms 14.3 Engineering applications of Laplace transforms: problems 14.3.1 Flywheel 14.3.2 RLC circuit 14.3.3 Problem: RC circuit 14.3.4 Problem: Newton's law of cooling 14.3.5 Problem: servo positioning system 14.3.6 Problem: robotic arm 15---Specific-mathematical-software-to-solve-_2020_Calculus-for-Engineering- 15 Specific mathematical software to solve some problems 15.1 Vibration and harmonic analysis 15.2 Critical forces - how to solve nonlinear equations and their systems 15.3 Shortest path problem and computer algorithms 15.4 Snails, snakes, and first-order ordinary differential equations 15.4.1 The epidemiological, and also malware propagation, model of Kermack and McKendrick 15.4.2 Coevolution and chirality: a story of snails and snakes 15.5 Oscillations in higher-order differential equations and systems of differential equations References Index_2020_Calculus-for-Engineering-Students Index