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''JUST THE MATHS''
by
A.J. Hobson
TEACHING UNITS - TABLE OF CONTENTS
(Average number of pages = 1038 ¸140 = 7.4 per unit)
All units are in presented as .PDF files
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UNIT 1.1 - ALGEBRA 1 - INTRODUCTION TO ALGEBRA
1.1.1 The Language of Algebra 1.1.2 The Laws of Algebra 1.1.3 Priorities in Calculations 1.1.4 Factors 1.1.5 Exercises 1.1.6 Answers to exercises (6 pages)
UNIT 1.2 - ALGEBRA 2 - NUMBERWORK 1.2.1 Types of number 1.2.2 Decimal numbers 1.2.3 Use of electronic calculators 1.2.4 Scientific notation 1.2.5 Percentages 1.2.6 Ratio 1.2.7 Exercises 1.2.8 Answers to exercises (8 pages)
UNIT 1.3 - ALGEBRA 3 - INDICES AND RADICALS (OR SURDS) 1.3.1 Indices 1.3.2 Radicals (or Surds) 1.3.3 Exercises 1.3.4 Answers to exercises (8 pages)
UNIT 1.4 - ALGEBRA 4 - LOGARITHMS 1.4.1 Common logarithms 1.4.2 Logarithms in general 1.4.3 Useful Results 1.4.4 Properties of logarithms 1.4.5 Natural logarithms 1.4.6 Graphs of logarithmic and exponential functions 1.4.7 Logarithmic scales 1.4.8 Exercises 1.4.9 Answers to exercises (10 pages)
UNIT 1.5 - ALGEBRA 5 - MANIPULATION OF ALGEBRAIC EXPRESSIONS 1.5.1 Simplification of expressions 1.5.2 Factorisation
1.5.3 Completing the square in a quadratic expression 1.5.4 Algebraic Fractions 1.5.5 Exercises 1.5.6 Answers to exercises (9 pages)
UNIT 1.6 - ALGEBRA 6 - FORMULAE AND ALGEBRAIC EQUATIONS 1.6.1 Transposition of formulae 1.6.2 Solution of linear equations 1.6.3 Solution of quadratic equations 1.6.4 Exercises 1.6.5 Answers to exercises (7 pages)
UNIT 1.7 - ALGEBRA 7 - SIMULTANEOUS LINEAR EQUATIONS 1.7.1 Two simultaneous linear equations in two unknowns 1.7.2 Three simultaneous linear equations in three unknowns 1.7.3 Ill-conditioned equations 1.7.4 Exercises 1.7.5 Answers to exercises (6 pages)
UNIT 1.8 - ALGEBRA 8 - POLYNOMIALS 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials 1.8.5 Exercises 1.8.6 Answers to exercises (8 pages)
UNIT 1.9 - ALGEBRA 9 - THE THEORY OF PARTIAL FRACTIONS 1.9.1 Introduction 1.9.2 Standard types of partial fraction problem 1.9.3 Exercises 1.9.4 Answers to exercises (7 pages)
UNIT 1.10 - ALGEBRA 10 - INEQUALITIES 1 1.10.1 Introduction 1.10.2 Algebraic rules for inequalities 1.10.3 Intervals 1.10.4 Exercises 1.10.5 Answers to exercises (5 pages)
UNIT 1.11 - ALGEBRA 11 - INEQUALITIES 2 1.11.1 Recap on modulus, absolute value or numerical value 1.11.2 Interval inequalities 1.11.3 Exercises 1.11.4 Answers to exercises (5 pages)
UNIT 2.1 - SERIES 1 - ELEMENTARY PROGRESSIONS AND SERIES 2.1.1 Arithmetic progressions 2.1.2 Arithmetic series 2.1.3 Geometric progressions 2.1.4 Geometric series 2.1.5 More general progressions and series 2.1.6 Exercises
3.5.4 Answers to exercises (8 pages)
UNIT 4.1 - HYPERBOLIC FUNCTIONS 1 - DEFINITIONS, GRAPHS AND IDENTITIES 4.1.1 Introduction 4.1.2 Definitions 4.1.3 Graphs of hyperbolic functions 4.1.4 Hyperbolic identities 4.1.5 Osborn's rule 4.1.6 Exercises 4.1.7 Answers to exercises (7 pages)
UNIT 4.2 - HYPERBOLIC FUNCTIONS 2 - INVERSE HYPERBOLIC FUNCTIONS 4.2.1 Introduction 4.2.2 The proofs of the standard formulae 4.2.3 Exercises 4.2.4 Answers to exercises (6 pages)
UNIT 5.1 - GEOMETRY 1 - CO-ORDINATES, DISTANCE AND GRADIENT 5.1.1 Co-ordinates 5.1.2 Relationship between polar & cartesian co-ordinates 5.1.3 The distance between two points 5.1.4 Gradient 5.1.5 Exercises 5.1.6 Answers to exercises (5 pages)
UNIT 5.2 - GEOMETRY 2 - THE STRAIGHT LINE 5.2.1 Preamble 5.2.2 Standard equations of a straight line 5.2.3 Perpendicular straight lines 5.2.4 Change of origin 5.2.5 Exercises 5.2.6 Answers to exercises (8 pages)
UNIT 5.3 - GEOMETRY 3 - STRAIGHT LINE LAWS 5.3.1 Introduction 5.3.2 Laws reducible to linear form 5.3.3 The use of logarithmic graph paper 5.3.4 Exercises 5.3.5 Answers to exercises (7 pages)
UNIT 5.4 - GEOMETRY 4 - ELEMENTARY LINEAR PROGRAMMING 5.4.1 Feasible Regions 5.4.2 Objective functions 5.4.3 Exercises 5.4.4 Answers to exercises (9 pages)
UNIT 5.5 - GEOMETRY 5 - CONIC SECTIONS (THE CIRCLE) 5.5.1 Introduction 5.5.2 Standard equations for a circle 5.5.3 Exercises 5.5.4 Answers to exercises (5 pages)
UNIT 5.6 - GEOMETRY 6 - CONIC SECTIONS (THE PARABOLA)
5.6.1 Introduction (the standard parabola) 5.6.2 Other forms of the equation of a parabola 5.6.3 Exercises 5.6.4 Answers to exercises (6 pages)
UNIT 5.7 - GEOMETRY 7 - CONIC SECTIONS (THE ELLIPSE) 5.7.1 Introduction (the standard ellipse) 5.7.2 A more general form for the equation of an ellipse 5.7.2 Exercises 5.7.3 Answers to exercises (4 pages)
UNIT 5.8 - GEOMETRY 8 - CONIC SECTIONS (THE HYPERBOLA) 5.8.1 Introduction (the standard hyperbola) 5.8.2 Asymptotes 5.8.3 More general forms for the equation of a hyperbola 5.8.4 The rectangular hyperbola 5.8.5 Exercises 5.8.6 Answers to exercises (8 pages)
UNIT 5.9 - GEOMETRY 9 - CURVE SKETCHING IN GENERAL 5.9.1 Symmetry 5.9.2 Intersections with the co-ordinate axes 5.9.3 Restrictions on the range of either variable 5.9.4 The form of the curve near the origin 5.9.5 Asymptotes 5.9.6 Exercises 5.9.7 Answers to exercises (10 pages)
UNIT 5.10 - GEOMETRY 10 - GRAPHICAL SOLUTIONS 5.10.1 The graphical solution of linear equations 5.10.2 The graphical solution of quadratic equations 5.10.3 The graphical solution of simultaneous equations 5.10.4 Exercises 5.10.5 Answers to exercises (7 pages)
UNIT 5.11 - GEOMETRY 11 - POLAR CURVES 5.11.1 Introduction 5.11.2 The use of polar graph paper 5.11.3 Exercises 5.11.4 Answers to exercises (10 pages)
UNIT 6.1 - COMPLEX NUMBERS 1 - DEFINITIONS AND ALGEBRA 6.1.1 The definition of a complex number 6.1.2 The algebra of complex numbers 6.1.3 Exercises 6.1.4 Answers to exercises (8 pages)
UNIT 6.2 - COMPLEX NUMBERS 2 - THE ARGAND DIAGRAM 6.2.1 Introduction 6.2.2 Graphical addition and subtraction 6.2.3 Multiplication by j 6.2.4 Modulus and argument 6.2.5 Exercises
UNIT 7.4 - DETERMINANTS 4 - HOMOGENEOUS LINEAR EQUATIONS
7.4.1 Trivial and non-trivial solutions 7.4.2 Exercises 7.4.3 Answers to exercises (7 pages)
UNIT 8.1 - VECTORS 1 - INTRODUCTION TO VECTOR ALGEBRA 8.1.1 Definitions 8.1.2 Addition and subtraction of vectors 8.1.3 Multiplication of a vector by a scalar 8.1.4 Laws of algebra obeyed by vectors 8.1.5 Vector proofs of geometrical results 8.1.6 Exercises 8.1.7 Answers to exercises (7 pages)
UNIT 8.2 - VECTORS 2 - VECTORS IN COMPONENT FORM 8.2.1 The components of a vector 8.2.2 The magnitude of a vector in component form 8.2.3 The sum and difference of vectors in component form 8.2.4 The direction cosines of a vector 8.2.5 Exercises 8.2.6 Answers to exercises (6 pages)
UNIT 8.3 - VECTORS 3 - MULTIPLICATION OF ONE VECTOR BY ANOTHER 8.3.1 The scalar product (or 'dot' product) 8.3.2 Deductions from the definition of dot product 8.3.3 The standard formula for dot product 8.3.4 The vector product (or 'cross' product) 8.3.5 Deductions from the definition of cross product 8.3.6 The standard formula for cross product 8.3.7 Exercises 8.3.8 Answers to exercises (8 pages)
UNIT 8.4 - VECTORS 4 - TRIPLE PRODUCTS 8.4.1 The triple scalar product 8.4.2 The triple vector product 8.4.3 Exercises 8.4.4 Answers to exercises (7 pages)
UNIT 8.5 - VECTORS 5 - VECTOR EQUATIONS OF STRAIGHT LINES 8.5.1 Introduction 8.5.2 The straight line passing through a given point and parallel to a given vector 8.5.3 The straight line passing through two given points 8.5.4 The perpendicular distance of a point from a straight line 8.5.5 The shortest distance between two parallel straight lines 8.5.6 The shortest distance between two skew straight lines 8.5.7 Exercises 8.5.8 Answers to exercises (14 pages)
UNIT 8.6 - VECTORS 6 - VECTOR EQUATIONS OF PLANES 8.6.1 The plane passing through a given point and perpendicular to a given vector 8.6.2 The plane passing through three given points 8.6.3 The point of intersection of a straight line and a plane 8.6.4 The line of intersection of two planes
8.6.5 The perpendicular distance of a point from a plane 8.6.6 Exercises 8.6.7 Answers to exercises (9 pages)
UNIT 9.1 - MATRICES 1 - DEFINITIONS AND ELEMENTARY MATRIX ALGEBRA 9.1.1 Introduction 9.1.2 Definitions 9.1.3 The algebra of matrices (part one) 9.1.4 Exercises 9.1.5 Answers to exercises (8 pages)
UNIT 9.2 - MATRICES 2 - FURTHER MATRIX ALGEBRA 9.2.1 Multiplication by a single number 9.2.2 The product of two matrices 9.2.3 The non-commutativity of matrix products 9.2.4 Multiplicative identity matrices 9.2.5 Exercises 9.2.6 Answers to exercises (6 pages)
UNIT 9.3 - MATRICES 3 - MATRIX INVERSION AND SIMULTANEOUS EQUATIONS 9.3.1 Introduction 9.3.2 Matrix representation of simultaneous linear equations 9.3.3 The definition of a multiplicative inverse 9.3.4 The formula for a multiplicative inverse 9.3.5 Exercises 9.3.6 Answers to exercises (11 pages)
UNIT 9.4 - MATRICES 4 - ROW OPERATIONS 9.4.1 Matrix inverses by row operations 9.4.2 Gaussian elimination (the elementary version) 9.4.3 Exercises 9.4.4 Answers to exercises (10 pages)
UNIT 9.5 - MATRICES 5 - CONSISTENCY AND RANK 9.5.1 The consistency of simultaneous linear equations 9.5.2 The row-echelon form of a matrix 9.5.3 The rank of a matrix 9.5.4 Exercises 9.5.5 Answers to exercises (9 pages)
UNIT 9.6 - MATRICES 6 - EIGENVALUES AND EIGENVECTORS 9.6.1 The statement of the problem 9.6.2 The solution of the problem 9.6.3 Exercises 9.6.4 Answers to exercises (9 pages)
UNIT 9.7 - MATRICES 7 - LINEARLY INDEPENDENT AND NORMALISED EIGENVECTORS 9.7.1 Linearly independent eigenvectors 9.7.2 Normalised eigenvectors 9.7.3 Exercises 9.7.4 Answers to exercises (5 pages)
UNIT 9.8 - MATRICES 8 - CHARACTERISTIC PROPERTIES AND SIMILARITY
10.5.3 Exercises 10.5.4 Answers to exercises (5 pages)
UNIT 10.6 - DIFFERENTIATION 6 - DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS 10.6.1 Summary of results 10.6.2 The derivative of an inverse sine 10.6.3 The derivative of an inverse cosine 10.6.4 The derivative of an inverse tangent 10.6.5 Exercises 10.6.6 Answers to exercises (7 pages)
UNIT 10.7 - DIFFERENTIATION 7 - DERIVATIVES OF INVERSE HYPERBOLIC FUNCTIONS 10.7.1 Summary of results 10.7.2 The derivative of an inverse hyperbolic sine 10.7.3 The derivative of an inverse hyperbolic cosine 10.7.4 The derivative of an inverse hyperbolic tangent 10.7.5 Exercises 10.7.6 Answers to exercises (7 pages)
UNIT 10.8 - DIFFERENTIATION 8 - HIGHER DERVIVATIVES 10.8.1 The theory 10.8.2 Exercises 10.8.3 Answers to exercises (4 pages)
UNIT 11.1 - DIFFERENTIATION APPLICATIONS 1 - TANGENTS AND NORMALS 11.1.1 Tangents 11.1.2 Normals 11.1.3 Exercises 11.1.4 Answers to exercises (5 pages)
UNIT 11.2 - DIFFERENTIATION APPLICATIONS 2 - LOCAL MAXIMA, LOCAL MINIMA AND POINTS OF INFLEXION 11.2.1 Introduction 11.2.2 Local maxima 11.2.3 Local minima 11.2.4 Points of inflexion 11.2.5 The location of stationary points and their nature 11.2.6 Exercises 11.2.7 Answers to exercises (14 pages)
UNIT 11.3 - DIFFERENTIATION APPLICATIONS 3 - CURVATURE 11.3.1 Introduction 11.3.2 Curvature in cartesian co-ordinates 11.3.3 Exercises 11.3.4 Answers to exercises (6 pages)
UNIT 11.4 - DIFFERENTIATION APPLICATIONS 4 - CIRCLE, RADIUS AND CENTRE OF CURVATURE 11.4.1 Introduction 11.4.2 Radius of curvature 11.4.3 Centre of curvature 11.4.4 Exercises 11.4.5 Answers to exercises (5 pages)
UNIT 11.5 - DIFFERENTIATION APPLICATIONS 5 - MACLAURIN'S AND TAYLOR'S SERIES
11.5.1 Maclaurin's series 11.5.2 Standard series 11.5.3 Taylor's series 11.5.4 Exercises 11.5.5 Answers to exercises (10 pages)
UNIT 11.6 - DIFFERENTIATION APPLICATIONS 6 - SMALL INCREMENTS AND SMALL ERRORS 11.6.1 Small increments 11.6.2 Small errors 11.6.3 Exercises 11.6.4 Answers to exercises (8 pages)
UNIT 12.1 - INTEGRATION 1 - ELEMENTARY INDEFINITE INTEGRALS 12.1.1 The definition of an integral 12.1.2 Elementary techniques of integration 12.1.3 Exercises 12.1.4 Answers to exercises (11 pages)
UNIT 12.2 - INTEGRATION 2 - INTRODUCTION TO DEFINITE INTEGRALS 12.2.1 Definition and examples 12.2.2 Exercises 12.2.3 Answers to exercises (3 pages)
UNIT 12.3 - INTEGRATION 3 - THE METHOD OF COMPLETING THE SQUARE 12.3.1 Introduction and examples 12.3.2 Exercises 12.3.3 Answers to exercises (4 pages)
UNIT 12.4 - INTEGRATION 4 - INTEGRATION BY SUBSTITUTION IN GENERAL 12.4.1 Examples using the standard formula 12.4.2 Integrals involving a function and its derivative 12.4.3 Exercises 12.4.4 Answers to exercises (5 pages)
UNIT 12.5 - INTEGRATION 5 - INTEGRATION BY PARTS 12.5.1 The standard formula 12.5.2 Exercises 12.5.3 Answers to exercises (6 pages)
UNIT 12.6 - INTEGRATION 6 - INTEGRATION BY PARTIAL FRACTIONS 12.6.1 Introduction and illustrations 12.6.2 Exercises 12.6.3 Answers to exercises (4 pages)
UNIT 12.7 - INTEGRATION 7 - FURTHER TRIGONOMETRIC FUNCTIONS 12.7.1 Products of sines and cosines 12.7.2 Powers of sines and cosines 12.7.3 Exercises 12.7.4 Answers to exercises (7 pages)
UNIT 12.8 - INTEGRATION 8 - THE TANGENT SUBSTITUTIONS 12.8.1 The substitution t = tanx 12.8.2 The substitution t = tan(x/2)
UNIT 13.7 - INTEGRATION APPLICATIONS 7 - FIRST MOMENTS OF AN AREA
13.7.1 Introduction 13.7.2 First moment of an area about the y-axis 13.7.3 First moment of an area about the x-axis 13.7.4 The centroid of an area 13.7.5 Exercises 13.7.6 Answers to exercises (12 pages)
UNIT 13.8 - INTEGRATION APPLICATIONS 8 - FIRST MOMENTS OF A VOLUME 13.8.1 Introduction 13.8.2 First moment of a volume of revolution about a plane through the origin, perpendicular to the x-axis 13.8.3 The centroid of a volume 13.8.4 Exercises 13.8.5 Answers to exercises (10 pages)
UNIT 13.9 - INTEGRATION APPLICATIONS 9 - FIRST MOMENTS OF A SURFACE OF REVOLUTION 13.9.1 Introduction 13.9.2 Integration formulae for first moments 13.9.3 The centroid of a surface of revolution 13.9.4 Exercises 13.9.5 Answers to exercises (11 pages)
UNIT 13.10 - INTEGRATION APPLICATIONS 10 - SECOND MOMENTS OF AN ARC 13.10.1 Introduction 13.10.2 The second moment of an arc about the y-axis 13.10.3 The second moment of an arc about the x-axis 13.10.4 The radius of gyration of an arc 13.10.5 Exercises 13.10.6 Answers to exercises (11 pages)
UNIT 13.11 - INTEGRATION APPLICATIONS 11 - SECOND MOMENTS OF AN AREA (A) 13.11.1 Introduction 13.11.2 The second moment of an area about the y-axis 13.11.3 The second moment of an area about the x-axis 13.11.4 Exercises 13.11.5 Answers to exercises (8 pages)
UNIT 13.12 - INTEGRATION APPLICATIONS 12 - SECOND MOMENTS OF AN AREA (B) 13.12.1 The parallel axis theorem 13.12.2 The perpendicular axis theorem 13.12.3 The radius of gyration of an area 13.12.4 Exercises 13.12.5 Answers to exercises (8 pages)
UNIT 13.13 - INTEGRATION APPLICATIONS 13 - SECOND MOMENTS OF A VOLUME (A) 13.13.1 Introduction 13.13.2 The second moment of a volume of revolution about the y-axis 13.13.3 The second moment of a volume of revolution about the x-axis 13.13.4 Exercises 13.13.5 Answers to exercises (8 pages)
UNIT 13.14 - INTEGRATION APPLICATIONS 14 - SECOND MOMENTS OF A VOLUME (B)
13.14.1 The parallel axis theorem 13.14.2 The radius of gyration of a volume 13.14.3 Exercises 13.14.4 Answers to exercises (6 pages)
UNIT 13.15 - INTEGRATION APPLICATIONS 15 - SECOND MOMENTS OF A SURFACE OF REVOLUTION 13.15.1 Introduction 13.15.2 Integration formulae for second moments 13.15.3 The radius of gyration of a surface of revolution 13.15.4 Exercises 13.15.5 Answers to exercises (9 pages)
UNIT 13.16 - INTEGRATION APPLICATIONS 16 - CENTRES OF PRESSURE 13.16.1 The pressure at a point in a liquid 13.16.2 The pressure on an immersed plate 13.16.3 The depth of the centre of pressure 13.16.4 Exercises 13.16.5 Answers to exercises (9 pages)
UNIT 14.1 - PARTIAL DIFFERENTIATION 1 - PARTIAL DERIVATIVES OF THE FIRST ORDER 14.1.1 Functions of several variables 14.1.2 The definition of a partial derivative 14.1.3 Exercises 14.1.4 Answers to exercises (7 pages)
UNIT 14.2 - PARTIAL DIFFERENTIATION 2 - PARTIAL DERIVATIVES OF THE SECOND AND HIGHER ORDERS 14.2.1 Standard notations and their meanings 14.2.2 Exercises 14.2.3 Answers to exercises (5 pages)
UNIT 14.3 - PARTIAL DIFFERENTIATION 3 - SMALL INCREMENTS AND SMALL ERRORS 14.3.1 Functions of one independent variable - a recap 14.3.2 Functions of more than one independent variable 14.3.3 The logarithmic method 14.3.4 Exercises 14.3.5 Answers to exercises (10 pages)
UNIT 14.4 - PARTIAL DIFFERENTIATION 4 - EXACT DIFFERENTIALS 14.4.1 Total differentials 14.4.2 Testing for exact differentials 14.4.3 Integration of exact differentials 14.4.4 Exercises 14.4.5 Answers to exercises (9 pages)
UNIT 14.5 - PARTIAL DIFFERENTIATION 5 - PARTIAL DERIVATIVES OF COMPOSITE FUNCTIONS 14.5.1 Single independent variables 14.5.2 Several independent variables 14.5.3 Exercises 14.5.4 Answers to exercises (8 pages)
15.3.3 Exercises 15.3.4 Answers to exercises (9 pages)
UNIT 15.4 - ORDINARY DIFFERENTIAL EQUATIONS 4 - SECOND ORDER EQUATIONS (A) 15.4.1 Introduction 15.4.2 Second order homogeneous equations 15.4.3 Special cases of the auxiliary equation 15.4.4 Exercises 15.4.5 Answers to exercises (9 pages)
UNIT 15.5 - ORDINARY DIFFERENTIAL EQUATIONS 5 - SECOND ORDER EQUATIONS (B) 15.5.1 Non-homogeneous differential equations 15.5.2 Determination of simple particular integrals 15.5.3 Exercises 15.5.4 Answers to exercises (6 pages)
UNIT 15.6 - ORDINARY DIFFERENTIAL EQUATIONS 6 - SECOND ORDER EQUATIONS (C) 15.6.1 Recap 15.6.2 Further types of particular integral 15.6.3 Exercises 15.6.4 Answers to exercises (7 pages)
UNIT 15.7 - ORDINARY DIFFERENTIAL EQUATIONS 7 - SECOND ORDER EQUATIONS (D) 15.7.1 Problematic cases of particular integrals 15.7.2 Exercises 15.7.3 Answers to exercises (6 pages)
UNIT 15.8 - ORDINARY DIFFERENTIAL EQUATIONS 8 - SIMULTANEOUS EQUATIONS (A) 15.8.1 The substitution method 15.8.2 Exercises 15.8.3 Answers to exercises (5 pages)
UNIT 15.9 - ORDINARY DIFFERENTIAL EQUATIONS 9 - SIMULTANEOUS EQUATIONS (B) 15.9.1 Introduction 15.9.2 Matrix methods for homogeneous systems 15.9.3 Exercises 15.9.4 Answers to exercises (8 pages)
UNIT 15.10 - ORDINARY DIFFERENTIAL EQUATIONS 10 - SIMULTANEOUS EQUATIONS (C) 15.10.1 Matrix methods for non-homogeneous systems 15.10.2 Exercises 15.10.3 Answers to exercises (10 pages)
UNIT 16.1 - LAPLACE TRANSFORMS 1 - DEFINITIONS AND RULES 16.1.1 Introduction 16.1.2 Laplace Transforms of simple functions 16.1.3 Elementary Laplace Transform rules 16.1.4 Further Laplace Transform rules 16.1.5 Exercises 16.1.6 Answers to exercises (10 pages)
UNIT 16.2 - LAPLACE TRANSFORMS 2 - INVERSE LAPLACE TRANSFORMS 16.2.1 The definition of an inverse Laplace Transform 16.2.2 Methods of determining an inverse Laplace Transform
16.2.3 Exercises 16.2.4 Answers to exercises (8 pages)
UNIT 16.3 - LAPLACE TRANSFORMS 3 - DIFFERENTIAL EQUATIONS 16.3.1 Examples of solving differential equations 16.3.2 The general solution of a differential equation 16.3.3 Exercises 16.3.4 Answers to exercises (7 pages)
UNIT 16.4 - LAPLACE TRANSFORMS 4 - SIMULTANEOUS DIFFERENTIAL EQUATIONS 16.4.1 An example of solving simultaneous linear differential equations 16.4.2 Exercises 16.4.3 Answers to exercises (5 pages)
UNIT 16.5 - LAPLACE TRANSFORMS 5 - THE HEAVISIDE STEP FUNCTION 16.5.1 The definition of the Heaviside step function 16.5.2 The Laplace Transform of H(t - T) 16.5.3 Pulse functions 16.5.4 The second shifting theorem 16.5.5 Exercises 16.5.6 Answers to exercises (8 pages)
UNIT 16.6 - LAPLACE TRANSFORMS 6 - THE DIRAC UNIT IMPULSE FUNCTION 16.6.1 The definition of the Dirac unit impulse function 16.6.2 The Laplace Transform of the Dirac unit impulse function 16.6.3 Transfer functions 16.6.4 Steady-state response to a single frequency input 16.6.5 Exercises 16.6.6 Answers to exercises (11 pages)
UNIT 16.7 - LAPLACE TRANSFORMS 7 - (AN APPENDIX) One view of how Laplace Transforms might have arisen (4 pages)
UNIT 16.8 - Z-TRANSFORMS 1 - DEFINITION AND RULES 16.8.1 Introduction 16.8.2 Standard Z-Transform definition and results 16.8.3 Properties of Z-Transforms 16.8.4 Exercises 16.8.5 Answers to exercises (10 pages)
UNIT 16.9 - Z-TRANSFORMS 2 - INVERSE Z-TRANSFORMS 16.9.1 The use of partial fractions 16.9.2 Exercises 16.9.3 Answers to exercises (6 pages)
UNIT 16.10 - Z-TRANSFORMS 3 - SOLUTION OF LINEAR DIFFERENCE EQUATIONS 16.10.1 First order linear difference equations 16.10.2 Second order linear difference equations 16.10.3 Exercises 16.10.4 Answers to exercises (9 pages)
UNIT 17.1 - NUMERICAL MATHEMATICS 1 - THE APPROXIMATE SOLUTION OF ALGEBRAIC EQUATIONS 17.1.1 Introduction
18.1.5 Selected answers to exercises (8 pages)
UNIT 18.2 - STATISTICS 2 - MEASURES OF CENTRAL TENDENCY 18.2.1 Introduction 18.2.2 The arithmetic mean (by coding) 18.2.3 The median 18.2.4 The mode 18.2.5 Quantiles 18.2.6 Exercises 18.2.7 Answers to exercises (9 pages)
UNIT 18.3 - STATISTICS 3 - MEASURES OF DISPERSION (OR SCATTER) 18.3.1 Introduction 18.3.2 The mean deviation 18.3.3 Practical calculation of the mean deviation 18.3.4 The root mean square (or standard) deviation 18.3.5 Practical calculation of the standard deviation 18.3.6 Other measures of dispersion 18.3.7 Exercises 18.3.8 Answers to exercises (6 pages)
UNIT 18.4 - STATISTICS 4 - THE PRINCIPLE OF LEAST SQUARES 18.4.1 The normal equations 18.4.2 Simplified calculation of regression lines 18.4.3 Exercises 18.4.4 Answers to exercises (6 pages)
UNIT 19.1 - PROBABILITY 1 - DEFINITIONS AND RULES 19.1.1 Introduction 19.1.2 Application of probability to games of chance 19.1.3 Empirical probability 19.1.4 Types of event 19.1.5 Rules of probability 19.1.6 Conditional probabilities 19.1.7 Exercises 19.1.8 Answers to exercises (5 pages)
UNIT 19.2 - PROBABILITY 2 - PERMUTATIONS AND COMBINATIONS 19.2.1 Introduction 19.2.2 Rules of permutations and combinations 19.2.3 Permutations of sets with some objects alike 19.2.4 Exercises 19.2.5 Answers to exercises (7 pages)
UNIT 19.3 - PROBABILITY 3 - RANDOM VARIABLES 19.3.1 Defining random variables 19.3.2 Probability distribution and probability density functions 19.3.3 Exercises 19.3.4 Answers to exercises (9 pages)
UNIT 19.4 - PROBABILITY 4 - MEASURES OF LOCATION AND DISPERSION 19.4.1 Common types of measure
19.4.2 Exercises 19.4.3 Answers to exercises (6 pages)
UNIT 19.5 - PROBABILITY 5 - THE BINOMIAL DISTRIBUTION 19.5.1 Introduction and theory 19.5.2 Exercises 19.5.3 Answers to exercises (5 pages)
UNIT 19.6 - PROBABILITY 6 - STATISTICS FOR THE BINOMIAL DISTRIBUTION 19.6.1 Construction of histograms 19.6.2 Mean and standard deviation of a binomial distribution 19.6.3 Exercises 19.6.4 Answers to exercises (10 pages)
UNIT 19.7 - PROBABILITY 7 - THE POISSON DISTRIBUTION 19.7.1 The theory 19.7.2 Exercises 19.7.3 Answers to exercises (5 pages)
UNIT 19.8 - PROBABILITY 8 - THE NORMAL DISTRIBUTION 19.8.1 Limiting position of a frequency polygon 19.8.2 Area under the normal curve 19.8.3 Normal distribution for continuous variables 19.8.4 Exercises 19.8.5 Answers to exercises (10 pages)
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