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

✍ Scribed by Jesus Martin Vaquero (editor), Michael Carr (editor), Araceli Queiruga Dios (editor), Daniela Richtarikova (editor)

• Publisher: Academic Press
• Year: 2020
• Tongue: English
• Leaves: 370
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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