Fundamental Engineering Mathematics

Neil Challis was born in Cambridge, UK. He studied mathematics at the University of Bristol and subsequently worked for some years as a mathematician in the British Gas Engineering Research Station at Killingworth. Since 1977, he has worked in the Mathematics Group, Sheffield Hallam University, UK and is currently head of that group. He obtained a PhD in mathematics from the University of Sheffield in 1988 and has taught mathematics to a wide variety of students, across the spectrum from first year engineers and other non-mathematicians who need access to mathematical ideas, techniques and thinking, to final year single honours mathematics students. Harry Gretton was born in Leicester, UK. He studied mathematics at the University of Sheffield, obtaining his PhD from there in 1970. He has taught mathematics sciences since then, both at Sheffield University and Sheffield Hallam University, and has been a tutor with the Open University since it was conceived. He has taught many varied students on many varied mathematically-related courses. In recent years he has developed a particular interest in the impact of technology on the way mathematics is taught, practiced and assessed.

About the Authors
Foreword
1: Numbers, Graphics and Algebra

1.1 NUMBERS, GRAPHICS AND ALGEBRA
1.2 WHAT NUMBERS ARE
1.3 HOW NUMBERS (AND LETTERS) BEHAVE
1.4 FRACTIONS, DECIMALS AND SCIENTIFIC NOTATION
1.5 POWERS OR INDICES
1.6 ANGLE AND LENGTH - GEOMETRY AND TRIGONOMETRY
END OF CHAPTER 1 - CALCULATOR ACTIVITIES – DO THESE NOW!


2: Linking Algebra and Graphics 1

2.1 ALGEBRA AND PICTURES
2.2 NUMBERS, LETTERS AND BRACKETS
2.3 “SPEAKING” ALGEBRA
2.4 ALGEBRAIC FRACTIONS
2.5 SOLVING SIMPLE EQUATIONS
2.6 CONNECTING STRAIGHT LINES AND LINEAR EXPRESSIONS
2.7 SOLVING LINEAR EQUATIONS GRAPHICALLY
2.8 TRANSPOSING FORMULAE
2.9 STRAIGHT LINES IN ENGINEERING
2.10 STRATEGIES FOR HANDLING LINEAR EQUATIONS AND GRAPHS
END OF CHAPTER 2 - MIXED ACTIVITIES – DO THESE NOW!


3: Linking Algebra and Graphics 2

3.1 MORE ON CONNECTING ALGEBRA TO GRAPHS
3.2 QUADRATIC FUNCTIONS
3.3 SOLVING QUADRATIC EQUATIONS
3.4 AN ALGEBRAIC TRICK - COMPLETING THE SQUARE
3.5 A DIVERSION - MATCH THE GRAPHS WITH THE FUNCTIONS
3.6 STRATEGIES FOR HANDLING QUADRATIC FUNCTIONS
3.7 WHERE NEXT WITH POLYNOMIALS?
END OF CHAPTER 3 ACTIVITY - DO THIS NOW!


4: Other Essential Functions

4.1 ESSENTIAL ENGINEERING FUNCTIONS
4.2 THE BASICS OF EXPONENTIALS AND LOGARITHMS
4.3 HOW THE EXPONENTIAL FUNCTION BERAYES
4.4 HOW THE LOGARITHM FUNCTION BEHAVES
4.5 THE BASICS OF TRIGONOMETRIC FUNCTIONS
4.6 INVERSE FUNCTIONS AND TRIGONOMETRIC EQUATIONS
END OF CHAPTER 4 - MIXED ACTIVITIES!


5: Combining and Applying Mathematical Tools

5.1 USING YOUR TOOLBOX
5.2 THE MOST BASIC FUNCTION – THE STRAIGHT LINE
5.3 TRANSFORMATIONS OF GRAPHS
5.4 DECAYING OSCILLATIONS
5.5 A FOGGY FUNCTION
5.6 HEAT LOSS IN BUILDINGS – A MATHEMATICAL MODEL


6: Complex Numbers

6.1 THE NEED FOR COMPLEX NUMBERS
6.2 THE j NOTATION AND COMPLEX NUMBERS
6.3 ARITHMETIC WITH COMPLEX NUMBERS
6.4 GEOMETRY WITH COMPLEX NUMBERS: THE ARGAND DIAGRAM
6.5 CARTESIAN AND POLAR FORM, MODULUS AND ARGUMENT
6.6 EULER’S RELATIONSHIP AND EXPONENTIAL FORM
6.7 SOME USES OF POLAR AND EXPONENTIAL FORM
6.8 COMPLEX ALGEBRA
6.9 ROOTS OF COMPLEX NUMBERS
6.10 MINI CASE STUDY
END OF CHAPTER 6 – MIXED EXERCISES – DO ALL THESE NOW!


7: Differential Calculus 1

7.1 THE NEED FOR DIFFERENTIAL CALCULUS
7.2 DIFFERENTIAL CALCULUS IN USE
7.3 WHAT DIFFERENTIATION MEANS GRAPHICALLY
7.4 VARIOUS WAYS OF FINDING DERIVATIVES
7.5 NUMERICAL DIFFERENTIATION
7.6 PAPER AND PENCIL APPROACHES TO DIFFERENTIATION
7.7 COMPUTER ALGEBRA SYSTEMS OR SYMBOL MANIPULATORS


8: Differential Calculus 2

8.1 DIFFERENTIAL CALCULUS: TAKING THE IDEAS FURTHER
8.2 SOLVING THE EXAMPLES FROM CHAPTER 7
8.3 HIGHER ORDER DERIVATIVES AND THEIR MEANING
8.4 FINDING MAXIMUM AND MINIMUM POINTS
8.5 PARAMETRIC DIFFERENTIATION
8.6 IMPLICIT DIFFERENTIATION
8.7 PARTIAL DIFFERENTIATION
8.8 AN ENGINEERING CASE STUDY
END OF CHAPTER 8 EXERCISES – DO ALL THESE NOW!


9: Integral Calculus 1

9.1 THE NEED FOR INTEGRAL CALCULUS
9.2 INDEFINITE INTEGRATION AND THE ARBITRARY CONSTANT
9.3 USING A COMPUTER ALGEBRA SYSTEM TO MAKE A TABLE OF INTEGRALS
9.4 DEFINITE INTEGRATION AND AREAS
9.5 USING AREAS TO ESTIMATE INTEGRALS
9.6 APPROXIMATE INTEGRATION - THE TRAPEZIUM RULE AND SIMPSON’S RULE
9.7 PAPER AND PENCIL APPROACHES TO INTEGRATION
END OF CHAPTER 9 - MIXED EXERCISES – DO ALL THESE NOW!


10: Integral Calculus 2

10.1 INTRODUCTION
10.2 INTEGRATION AS SUMMATION: MEAN AND RMS
10.3 INTEGRATION AS SUMMATION: CHARGE ACCUMULATION
10.4 INTEGRATION AS SUMMATION: VOLUME AND SURFACE AREA
10.5 A FIRST LOOK AT DIFFERENTIAL EQUATIONS
END OF CHAPTER 10 EXERCISES - DO ALL OF THESE NOW!


11: Linear Simultaneous Equations

11.1 THE NEED FOR LINEAR SIMULTANEOUS EQUATIONS
11.2 WHERE SIMULTANEOUS EQUATIONS OCCUR – TWO EXAMPLES
11.3 SOLVING SIMULTANEOUS EQUATIONS GRAPHICALLY
11.4 SOLVING SIMULTANEOUS EQUATIONS WITH SIMPLE NUMBERS
11.5 SOLVING SIMULTANEOUS EQUATIONS ALGEBRAICALLY
11.6 SOLVING SIMULTANEOUS EQUATIONS USING TECHNOLOGY
11.7 EQUATIONS WITH NO UNIQUE SOLUTION - SINGULAR EQUATIONS
11.8 ILL CONDITIONED EQUATIONS
11.9 SOLVING THE “REAL” PROBLEMS
END OF CHAPTER 11 - MIXED EXERCISES – DO THESE NOW!


12: Matrices

12.1 MATRICES: WHAT ARE THEY, AND WHY DO YOU NEED THEM?
12.2 ARITHMETIC AND ALGEBRAIC OPERATIONS WITH MATRICES
12.3 MATRICES AND TECHNOLOGY
12.4 MATRICES AND SIMULTANEOUS EQUATIONS – THE MATRIX INVERSE
12.5 MATRICES AND GEOMETRICAL TRANSFORMATIONS
END OF CHAPTER 12 - MIXED EXERCISES - DO ALL THESE NOW!


13: More Linear Simultaneous Equations

13.1 LARGER SETS OF SIMULTANEOUS EQUATIONS
13.2 ELIMINATION METHODS: GAUSSIAN ELIMINATION
13.3 ITERATIVE METHODS
13.4 THE GAUSS-JORDAN METHOD
13.5 FINDING A MATRIX INVERSE BY THE GAUSS-JORDAN METHOD
13.6 ENGINEERING CASE STUDY: HEATING AND COOLING


14: Vectors

14.1 INTRODUCTION
14.2 REPRESENTING VECTORS
14.3 THE ALGEBRA OF VECTORS
14.4 PRODUCTS OF VECTORS
END OF CHAPTER 14 EXERCISES – DO ALL OF THESE NOW!


15: First Order Ordinary Differential Equations

15.1 INTRODUCTION
15.2 OVERVIEW
15.3 DIRECT INTEGRATION REVISITED
15.4 SOLUTION BY SEPARATION OF VARIABLES
15.5 ENGINEERING CASE STUDIES
15.6 NUMERICAL SOLUTION METHODS – THE EULER METHOD
15.7 EXPLORING THE PARAMETERS
END OF CHAPTER 15 MIXED EXERCISES – DO ALL OF THESE NOW!


16: Second Order Ordinary Differential Equations

16.1 INTRODUCTION
16.2 SECOND ORDER DIFFERENTIAL EQUATIONS
16.3 CASE STUDY: A SUSPENSION SYSTEM
16.4 THE SOLUTION OF LINEAR SECOND ORDER O.D.E.S
16.5 THE COMPLEMENTARY FUNCTION/PARTICULAR INTEGRAL APPROACH
16.6 THE CF - FINDING OUT ABOUT UNFORCED CHANGE
16.7 FINDING THE ARBITRARY CONSTANTS BY USING INITIAL CONDITIONS
16.8 FINDING THE P.I. - THE EFFECT OF FORCING CHANGE
16.9 TECHNOLOGICAL SOLVERS
16.10 SOME FINAL EXAMPLES
END OF CHAPTER 16 MIXED EXERCISES - DO ALL OF THESE NOW!


17: Laplace Transforms And Ordinary Differential Equations

17.1 THE USEFULNESS OF THE LAPLACE TRANSFORM
17.2 WHAT IS THE LAPLACE TRANSFORM?
17.3 THE LAPLACE TRANSFORM IN ACTION: FIRST ORDER ODES
17.4 USING THE LAPLACE TRANSFORM WITH SECOND ORDER ODEs
17.5 A FINAL SPECIAL CASE – RESONANCE
END OF CHAPTER 17 EXERCISES – DO ALL THESE NOW!


18: Taylor Series

18.1 THE ESSENTIAL ROLE OF TAYLOR SERIES
18.2 LINEARISATION
18.3 MACLAURIN SERIES
18.4 GETTING AWAY FROM x = 0: TAYLOR SERIES
18.5 USES OF TAYLOR AND MACLAURIN SERIES
END OF CHAPTER 18 PROBLEMS – DO ALL THESE NOW!


19: Statistics And Data Handling

19.1 WHY ENGINEERS NEED DATA HANDLING SKILLS
19.2 PRESENTING DATA IN PICTURES
19.3 SUMMARISING DATA SETS IN A FEW NUMBERS
19.4 FITTING LAWS TO EXPERIMENTAL DATA


20: Probability

20.1 WHAT IS PROBABILITY?
20.2 SIMPLE EXAMPLES – COMPLETE ENUMERATION
20.3 MORE COMPLEX SITVATIONS – THE LAWS OF PROBABILITY
20.4 TREE DIAGRAMS
20.5 SOME MORE PROBABILITY PROBLEMS
20.6 WHERE NEXT WITH PROBABILITY?


Glossary

G1 GREEK ALPHABET
G2 SI UNITS
G3 COMMON GRAPHS TO NOTE
G4 POWER SERIES
G5 COMMON NOTATION
G6 TABLE OF TRIGONOMETRIC FUNCTION FORMULAE