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Mathematics for Electrical Engineering and Computing

Book Description

Mathematics for Electrical Engineering and Computing embraces many applications of modern mathematics, such as Boolean Algebra and Sets and Functions, and also teaches both discrete and continuous systems - particularly vital for Digital Signal Processing (DSP). In addition, as most modern engineers are required to study software, material suitable for Software Engineering - set theory, predicate and prepositional calculus, language and graph theory - is fully integrated into the book.

Excessive technical detail and language are avoided, recognising that the real requirement for practising engineers is the need to understand the applications of mathematics in everyday engineering contexts. Emphasis is given to an appreciation of the fundamental concepts behind the mathematics, for problem solving and undertaking critical analysis of results, whether using a calculator or a computer.

The text is backed up by numerous exercises and worked examples throughout, firmly rooted in engineering practice, ensuring that all mathematical theory introduced is directly relevant to real-world engineering. The book includes introductions to advanced topics such as Fourier analysis, vector calculus and random processes, also making this a suitable introductory text for second year undergraduates of electrical, electronic and computer engineering, undertaking engineering mathematics courses.

The book is supported with a number of free online resources. On the companion website readers will find:
* over 60 pages of "Background Mathematics" reinforcing introductory material for revision purposes in advance of your first year course
* plotXpose software (for equation solving, and drawing graphs of simple functions, their derivatives, integrals and Fourier transforms)
* problems and projects (linking directly to the software)

In addition, for lecturers only, http://textbooks.elsevier.com features a complete worked solutions manual for the exercises in the book.

Dr Attenborough is a former Senior Lecturer in the School of Electrical, Electronic and Information Engineering at South Bank University. She is currently Technical Director of The Webbery - Internet development company, Co. Donegal, Ireland.

* Fundamental principles of mathematics introduced and applied in engineering practice, reinforced through over 300 examples directly relevant to real-world engineering
* Over 60 pages of basic revision material available to download in advance of embarking on a first year course
* Free website support, featuring complete solutions manual, background mathematics, plotXpose software, and further problems and projects enabling students to build on the concepts introduced, and put the theory into practice

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright page
  5. Preface
  6. Acknowledgements
  7. Part 1: Sets, functions, and calculus
    1. 1. Sets and functions
      1. 1.1 Introduction
      2. 1.2 Sets
      3. 1.3 Operations on sets
      4. 1.4 Relations and functions
      5. 1.5 Combining functions
      6. 1.6 Summary
      7. 1.7 Exercises
    2. 2. Functions and their graphs
      1. 2.1 Introduction
      2. 2.2 The straight line: <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">y</span> = = <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">mx</span> + + <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">c</span>
      3. 2.3 The quadratic function: <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">y</span> = = <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">ax</span><sup xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">2</sup>++<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">bx</span>++<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">c</span>
      4. 2.4 The function <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">y</span> = 1/ = 1/<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">x</span>
      5. 2.5 The functions <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">y</span> = = <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">a<sup>x</sup></span>
      6. 2.6 Graph sketching using simple transformations
      7. 2.7 The modulus function, <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">y</span> = | = |<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">x</span>| or | or <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">y</span> = a = a<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">bs</span>((<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">x</span>))
      8. 2.8 Symmetry of functions and their graphs
      9. 2.9 Solving inequalities
      10. 2.10 Using graphs to find an expression for the function from experimental data
      11. 2.11 Summary
      12. 2.12 Exercises
    3. 3. Problem solving and the art of the convincing argument
      1. 3.1 Introduction
      2. 3.2 Describing a problem in mathematical language
      3. 3.3 Propositions and predicates
      4. 3.4 Operations on propositions and predicates
      5. 3.5 Equivalence
      6. 3.6 Implication
      7. 3.7 Making sweeping statements
      8. 3.8 Other applications of predicates
      9. 3.9 Summary
      10. 3.10 Exercises
    4. 4. Boolean algebra
      1. 4.1 Introduction
      2. 4.2 Algebra
      3. 4.3 Boolean algebras
      4. 4.4 Digital circuits
      5. 4.5 Summary
      6. 4.6 Exercises
    5. 5. Trigonometric functions and waves
      1. 5.1 Introduction
      2. 5.2 Trigonometric functions and radians
      3. 5.3 Graphs and important properties
      4. 5.4 Wave functions of time and distance
      5. 5.5 Trigonometric identities
      6. 5.6 Superposition
      7. 5.7 Inverse trigonometric functions
      8. 5.8 Solving the trigonometric equations sin <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">x</span> = = <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">a,</span> cos cos <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">x= a,</span> tan tan <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">x</span> = = <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">a</span>
      9. 5.9 Summary
      10. 5.10 Exercises
    6. 6. Differentiation
      1. 6.1 Introduction
      2. 6.2 The average rate of change and the gradient of a chord
      3. 6.3 The derivative function
      4. 6.4 Some common derivatives
      5. 6.5 Finding the derivative of combinations of functions
      6. 6.6 Applications of differentiation
      7. 6.7 Summary
      8. 6.9 Exercises
    7. 7. Integration
      1. 7.1 Introduction
      2. 7.2 Integration
      3. 7.3 Finding integrals
      4. 7.4 Applications of integration
      5. 7.5 The definite integral
      6. 7.6 The mean value and r.m.s. value
      7. 7.7 Numerical Methods of Integration
      8. 7.8 Summary
      9. 7.9 Exercises
    8. 8. The exponential function
      1. 8.1 Introduction
      2. 8.2 Exponential growth and decay
      3. 8.3 The exponential function <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">y = e<sup>t</sup></span>
      4. 8.4 The hyperbolic functions
      5. 8.5 More differentiation and integration
      6. 8.6 Summary
      7. 8.7 Exercises
    9. 9. Vectors
      1. 9.1 Introduction
      2. 9.2 Vectors and vector quantities
      3. 9.3 Addition and subtraction of vectors
      4. 9.4 Magnitude and direction of a 2D vector–polar co-ordinates
      5. 9.5 Application of vectors to represent waves (phasors)
      6. 9.6 Multiplication of a vector by a scalar and unit vectors
      7. 9.7 Basis vectors
      8. 9.8 Products of vectors
      9. 9.9 Vector equation of a line
      10. 9.10 Summary
      11. 9.12 Exercises
    10. 10. Complex numbers
      1. 10.1 Introduction
      2. 10.2 Phasor rotation by π/2
      3. 10.3 Complex numbers and operations
      4. 10.4 Solution of quadratic equations
      5. 10.5 Polar form of a complex number
      6. 10.6 Applications of complex numbers to AC linear circuits
      7. 10.7 Circular motion
      8. 10.8 The importance of being exponential
      9. 10.9 Summary
      10. 10.10 Exercises
    11. 11. Maxima and minima and sketching functions
      1. 11.1 Introduction
      2. 11.2 Stationary points, local maxima and minima
      3. 11.3 Graph sketching by analysing the function behaviour
      4. 11.4 Summary
      5. 11.5 Exercises
    12. 12. Sequences and series
      1. 12.1 Introduction
      2. 12.2 Sequences and series definitions
      3. 12.3 Arithmetic progression
      4. 12.4 Geometric progression
      5. 12.5 Pascal's triangle and the binomial series
      6. 12.6 Power series
      7. 12.7 Limits and convergence
      8. 12.8 Newton–Raphson method for solving equations
      9. 12.9 Summary
      10. 12.10 Exercises
  8. Part 2: Systems
    1. 13. Systems of linear equations, matrices, and determinants
      1. 13.1 Introduction
      2. 13.2 Matrices
      3. 13.3 Transformations
      4. 13.4 Systems of equations
      5. 13.5 Gauss elimination
      6. 13.6 The inverse and determinant of a 3 × 3 matrix
      7. 13.7 Eigenvectors and eigenvalues
      8. 13.8 Least squares data fitting
      9. 13.9 Summary
      10. 13.10 Exercises
    2. 14. Differential equations and difference equations
      1. 14.1 Introduction
      2. 14.2 Modelling simple systems
      3. 14.3 Ordinary differential equations
      4. 14.4 Solving first-order LTI systems
      5. 14.5 Solution of a second-order LTI systems
      6. 14.6 Solving systems of differential equations
      7. 14.7 Difference equations
      8. 14.8 Summary
      9. 14.9 Exercises
    3. 15. Laplace and <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">z</span> transforms transforms
      1. 15.1 Introduction
      2. 15.2 The Laplace transform – definition
      3. 15.3 The unit step function and the (impulse) delta function
      4. 15.4 Laplace transforms of simple functions and properties of the transform
      5. 15.5 Solving linear differential equations with constant coefficients
      6. 15.6 Laplace transforms and systems theory
      7. 15.7 <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">z</span> transforms transforms
      8. 15.8 Solving linear difference equations with constant coefficients using <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">z</span> transforms transforms
      9. 15.9 <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="italic">z</span> transforms and systems theory transforms and systems theory
      10. 15.10 Summary
      11. 15.11 Exercises
    4. 16. Fourier series
      1. 16.1 Introduction
      2. 16.2 Periodic Functions
      3. 16.3 Sine and cosine series
      4. 16.4 Fourier series of symmetric periodic functions
      5. 16.5 Amplitude and phase representation of a Fourier series
      6. 16.6 Fourier series in complex form
      7. 16.7 Summary
      8. 16.8 Exercises
  9. Part 3: Functions of more than one variable
    1. 17. Functions of more than one variable
      1. 17.1 Introduction
      2. 17.2 Functions of two variables – surfaces
      3. 17.3 Partial differentiation
      4. 17.4 Changing variables – the chain rule
      5. 17.5 The total derivative along a path
      6. 17.6 Higher-order partial derivatives
      7. 17.7 Summary
      8. 17.8 Exercises
    2. 18. Vector calculus
      1. 18.1 Introduction
      2. 18.2 The gradient of a scalar field
      3. 18.3 Differentiating vector fields
      4. 18.4 The scalar line integral
      5. 18.5 Surface integrals
      6. 18.6 Summary
      7. 18.7 Exercises
  10. Part 4: Graph and language theory
    1. 19. Graph theory
      1. 19.1 Introduction
      2. 19.2 Definitions
      3. 19.3 Matrix representation of a graph
      4. 19.4 Trees
      5. 19.5 The shortest path problem
      6. 19.6 Networks and maximum flow
      7. 19.7 State transition diagrams
      8. 19.8 Summary
      9. 19.9 Exercises
    2. 20. Language theory
      1. 20.1 Introduction
      2. 20.2 Languages and grammars
      3. 20.3 Derivations and derivation trees
      4. 20.4 Extended Backus-Naur Form (EBNF)
      5. 20.5 Extensible markup language (XML)
      6. 20.6 Summary
      7. 20.7 Exercises
  11. Part 5: Probability and statistics
    1. 21. Probability and statistics
      1. 21.1 Introduction
      2. 21.2 Population and sample, representation of data, mean, variance and standard deviation
      3. 21.3 Random systems and probability
      4. 21.4 Addition law of probability
      5. 21.5 Repeated trials, outcomes, and probabilities
      6. 21.6 Repeated trials and probability trees
      7. 21.7 Conditional probability and probability trees
      8. 21.8 Application of the probability laws to the probability of failure of an electrical circuit
      9. 21.9 Statistical modelling
      10. 21.10 The normal distribution
      11. 21.11 The exponential distribution
      12. 21.12 The binomial distribution
      13. 21.13 The Poisson distribution
      14. 21.14 Summary
      15. 21.15 Exercises
  12. Answers to exercises
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
    17. Chapter 17
    18. Chapter 18
    19. Chapter 19
    20. Chapter 20
    21. Chapter 21
  13. Index