# ''JUST THE MATHS''

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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
2.1.7 Answers to exercises (12 pages)

UNIT 2.2 - SERIES 2 - BINOMIAL SERIES
2.2.1 Pascal's Triangle
2.2.2 Binomial Formulae
2.2.3 Exercises
2.2.4 Answers to exercises (9 pages)

UNIT 2.3 - SERIES 3 - ELEMENTARY CONVERGENCE AND DIVERGENCE
2.3.1 The definitions of convergence and divergence
2.3.2 Tests for convergence and divergence (positive terms)
2.3.3 Exercises
2.3.4 Answers to exercises (13 pages)

UNIT 2.4 - SERIES 4 - FURTHER CONVERGENCE AND DIVERGENCE
2.4.1 Series of positive and negative terms
2.4.2 Absolute and conditional convergence
2.4.3 Tests for absolute convergence
2.4.4 Power series
2.4.5 Exercises
2.4.6 Answers to exercises (9 pages)

UNIT 3.1 - TRIGONOMETRY 1 - ANGLES AND TRIGONOMETRIC FUNCTIONS
3.1.1 Introduction
3.1.2 Angular measure
3.1.3 Trigonometric functions
3.1.4 Exercises
3.1.5 Answers to exercises (6 pages)

UNIT 3.2 - TRIGONOMETRY 2 - GRAPHS OF TRIGONOMETRIC FUNCTIONS
3.2.1 Graphs of elementary trigonometric functions
3.2.2 Graphs of more general trigonometric functions
3.2.3 Exercises
3.2.4 Answers to exercises (7 pages)

UNIT 3.3 - TRIGONOMETRY 3 - APPROXIMATIONS AND INVERSE
FUNCTIONS
3.3.1 Approximations for trigonometric functions
3.3.2 Inverse trigonometric functions
3.3.3 Exercises
3.3.4 Answers to exercises (6 pages)

UNIT 3.4 - TRIGONOMETRY 4 - SOLUTION OF TRIANGLES
3.4.1 Introduction
3.4.2 Right-angled triangles
3.4.3 The sine and cosine rules
3.4.4 Exercises
3.4.5 Answers to exercises (5 pages)

UNIT 3.5 - TRIGONOMETRY 5 - TRIGONOMETRIC IDENTITIES AND WAVE-FORMS
3.5.1 Trigonometric identities
3.5.2 Amplitude, wave-length, frequency and phase-angle
3.5.3 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
6.2.6 Answers to exercises (7 pages)

UNIT 6.3 - COMPLEX NUMBERS 3 - THE POLAR AND EXPONENTIAL FORMS
6.3.1 The polar form
6.3.2 The exponential form
6.3.3 Products and quotients in polar form
6.3.4 Exercises
6.3.5 Answers to exercises (8 pages)

UNIT 6.4 - COMPLEX NUMBERS 4 - POWERS OF COMPLEX NUMBERS
6.4.1 Positive whole number powers
6.4.2 Negative whole number powers
6.4.3 Fractional powers & De Moivre's Theorem
6.4.4 Exercises
6.4.5 Answers to exercises (5 pages)

UNIT 6.5 - COMPLEX NUMBERS 5 - APPLICATIONS TO TRIGONOMETRIC IDENTITIES
6.5.1 Introduction
6.5.2 Expressions for cosn q, sinn q in terms of cosq, sinq
6.5.3 Expressions for cosnq and sinnq in terms of sines and cosines of whole multiples of x
6.5.4 Exercises
6.5.5 Answers to exercises (5 pages)

UNIT 6.6 - COMPLEX NUMBERS 6 - COMPLEX LOCI
6.6.1 Introduction
6.6.2 The circle
6.6.3 The half-straight-line
6.6.4 More general loci
6.6.5 Exercises
6.6.6 Answers to exercises (6 pages)

UNIT 7.1 - DETERMINANTS 1 - SECOND ORDER DETERMINANTS
7.1.1 Pairs of simultaneous linear equations
7.1.2 The definition of a second order determinant
7.1.3 Cramer's Rule for two simultaneous linear equations
7.1.4 Exercises
7.1.5 Answers to exercises (7 pages)

UNIT 7.2 - DETERMINANTS 2 - CONSISTENCY AND THIRD ORDER DETERMINANTS
7.2.1 Consistency for three simultaneous linear equations in two unknowns
7.2.2 The definition of a third order determinant
7.2.3 The rule of Sarrus
7.2.4 Cramer's rule for three simultaneous linear equations in three unknowns
7.2.5 Exercises
7.2.6 Answers to exercises (10 pages)

UNIT 7.3 - DETERMINANTS 3 - FURTHER EVALUATION OF 3 X 3 DETERMINANTS
7.3.1 Expansion by any row or column
7.3.2 Row and column operations on determinants
7.3.3 Exercises
7.3.4 Answers to exercises (10 pages)

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 TRANSFORMATIONS
9.8.1 Properties of eigenvalues and eigenvectors
9.8.2 Similar matrices
9.8.3 Exercises
9.7.4 Answers to exercises (9 pages)

UNIT 9.9 - MATRICES 9 - MODAL AND SPECTRAL MATRICES
9.9.1 Assumptions and definitions
9.9.2 Diagonalisation of a matrix
9.9.3 Exercises
9.9.4 Answers to exercises (9 pages)

UNIT 9.10 - MATRICES 10 - SYMMETRIC MATRICES AND QUADRATIC FORMS
9.10.1 Symmetric matrices
9.10.2 Quadratic forms
9.10.3 Exercises
9.10.4 Answers to exercises (7 pages)

UNIT 10.1 - DIFFERENTIATION 1 - FUNTIONS AND LIMITS
10.1.1 Functional notation
10.1.2 Numerical evaluation of functions
10.1.3 Functions of a linear function
10.1.4 Composite functions
10.1.5 Indeterminate forms
10.1.6 Even and odd functions
10.1.7 Exercises
10.1.8 Answers to exercises (12 pages)

UNIT 10.2 - DIFFERENTIATION 2 - RATES OF CHANGE
10.2.1 Introduction
10.2.2 Average rates of change
10.2.3 Instantaneous rates of change
10.2.4 Derivatives
10.2.5 Exercises
10.2.6 Answers to exercises (7 pages)

UNIT 10.3 - DIFFERENTIATION 3 - ELEMENTARY TECHNIQUES OF DIFFERENTIATION
10.3.1 Standard derivatives
10.3.2 Rules of differentiation
10.3.3 Exercises
10.3.4 Answers to exercises (9 pages)

UNIT 10.4 - DIFFERENTIATION 4 - PRODUCTS, QUOTIENTS AND LOGARITHMIC DIFFERENTIATION
10.4.1 Products
10.4.2 Quotients
10.4.3 Logarithmic differentiation
10.4.4 Exercises
10.4.5 Answers to exercises (10 pages)

UNIT 10.5 - DIFFERENTIATION 5 - IMPLICIT AND PARAMETRIC FUNCTIONS
10.5.1 Implicit functions
10.5.2 Parametric functions
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)
12.8.3 Exercises
12.8.4 Answers to exercises (5 pages)

UNIT 12.9 - INTEGRATION 9 - REDUCTION FORMULAE
12.9.1 Indefinite integrals
12.9.2 Definite integrals
12.9.3 Exercises
12.9.4 Answers to exercises (7 pages)

UNIT 12.10 - INTEGRATION 10 - FURTHER REDUCTION FORMULAE
12.10.1 Integer powers of a sine
12.10.2 Integer powers of a cosine
12.10.3 Wallis's formulae
12.10.4 Combinations of sines and cosines
12.10.5 Exercises
12.10.6 Answers to exercises (8 pages)

UNIT 13.1 - INTEGRATION APPLICATIONS 1 - THE AREA UNDER A CURVE
13.1.1 The elementary formula
13.1.2 Definite integration as a summation
13.1.3 Exercises
13.1.4 Answers to exercises (6 pages)

UNIT 13.2 - INTEGRATION APPLICATIONS 2 - MEAN AND ROOT MEAN SQUARE VALUES
13.2.1 Mean values
13.2.2 Root mean square values
13.2.3 Exercises
13.2.4 Answers to exercises (4 pages)

UNIT 13.3 - INTEGRATION APLICATIONS 3 - VOLUMES OF REVOLUTION
13.3.1 Volumes of revolution about the x-axis
13.3.2 Volumes of revolution about the y-axis
13.3.3 Exercises
13.3.4 Answers to exercises (7 pages)

UNIT 13.4 - INTEGRATION APPLICATIONS 4 - LENGTHS OF CURVES
13.4.1 The standard formulae
13.4.2 Exercises
13.4.3 Answers to exercises (5 pages)

UNIT 13.5 - INTEGRATION APPLICATIONS 5 - SURFACES OF REVOLUTION
13.5.1 Surfaces of revolution about the x-axis
13.5.2 Surfaces of revolution about the y-axis
13.5.3 Exercises
13.5.4 Answers to exercises (7 pages)

UNIT 13.6 - INTEGRATION APPLICATIONS 6 - FIRST MOMENTS OF AN ARC
13.6.1 Introduction
13.6.2 First moment of an arc about the y-axis
13.6.3 First moment of an arc about the x-axis
13.6.4 The centroid of an arc
13.6.5 Exercises
13.6.6 Answers to exercises (11 pages)

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)

UNIT 14.6 - PARTIAL DIFFERENTIATION 6 - IMPLICIT FUNCTIONS
14.6.1 Functions of two variables
14.6.2 Functions of three variables
14.6.3 Exercises
14.6.4 Answers to exercises (6 pages)

UNIT 14.7 - PARTIAL DIFFERENTIATON 7 - CHANGE OF INDEPENDENT VARIABLE
14.7.1 Illustrations of the method
14.7.2 Exercises
14.7.3 Answers to exercises (5 pages)

UNIT 14.8 - PARTIAL DIFFERENTIATON 8 - DEPENDENT AND INDEPENDENT FUNCTIONS
14.8.1 The Jacobian
14.8.2 Exercises
14.8.3 Answers to exercises (8 pages)

UNIT 14.9 - PARTIAL DIFFERENTIATON 9 - TAYLOR'S SERIES FOR FUNCTIONS OF SEVERAL VARIABLES
14.9.1 The theory and formula
14.9.2 Exercises (8 pages)

UNIT 14.10 - PARTIAL DIFFERENTIATON 10 - STATIONARY VALUES FOR FUNCTIONS OF TWO VARIABLES
14.10.1 Introduction
14.10.2 Sufficient conditions for maxima and minima 14.10.3 Exercises
14.10.4 Answers to exercises (9 pages)

UNIT 14.11 - PARTIAL DIFFERENTIATON 11 - CONSTRAINED MAXIMA AND MINIMA
14.11.1 The substitution method
14.11.2 The method of Lagrange multipliers 14.11.3 Exercises
14.11.4 Answers to exercises (11 pages)

UNIT 14.12 - PARTIAL DIFFERENTIATON 12 - THE PRINCIPLE OF LEAST SQUARES
14.12.1 The normal equations
14.11.2 Simplified calculation of regression lines 14.11.3 Exercises
14.11.4 Answers to exercises (9 pages)

UNIT 15.1 - ORDINARY DIFFERENTIAL EQUATIONS 1 - FIRST ORDER EQUATIONS (A)
15.1.1 Introduction and definitions
15.1.2 Exact equations
15.1.3 The method of separation of the variables
15.1.4 Exercises
15.1.5 Answers to exercises (8 pages)

UNIT 15.2 - ORDINARY DIFFERENTIAL EQUATIONS 2 - FIRST ORDER EQUATIONS (B)
15.2.1 Homogeneous equations
15.2.2 The standard method
15.2.3 Exercises
15.2.4 Answers to exercises (6 pages)

UNIT 15.3 - ORDINARY DIFFERENTIAL EQUATIONS 3 - FIRST ORDER EQUATIONS (C)
15.3.1 Linear equations
15.3.2 Bernouilli's equation
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
17.1.2 The Bisection method
17.1.3 The rule of false position
17.1.4 The Newton-Raphson method
17.1.5 Exercises
17.1.6 Answers to exercises (8 pages)

UNIT 17.2 - NUMERICAL MATHEMATICS 2 - APPROXIMATE INTEGRATION (A)
17.2.1 The trapezoidal rule
17.2.2 Exercises
17.2.3 Answers to exercises (4 pages)

UNIT 17.3 - NUMERICAL MATHEMATICS 3 - APPROXIMATE INTEGRATION (B)
17.3.1 Simpson's rule
17.3.2 Exercises
17.3.3 Answers to exercises (6 pages)

UNIT 17.4 - NUMERICAL MATHEMATICS 4 - FURTHER GAUSSIAN ELIMINATION
17.4.1 Gaussian elimination by ``partial pivoting''with a check column
17.4.2 Exercises
17.4.3 Answers to exercises (4 pages)

UNIT 17.5 - NUMERICAL MATHEMATICS 5 - ITERATIVE METHODS FOR SOLVING SIMULTANEOUS LINEAR EQUATIONS
17.5.1 Introduction
17.5.2 The Gauss-Jacobi iteration
17.5.3 The Gauss-Seidel iteration
17.5.4 Exercises
17.5.5 Answers to exercises (7 pages)

UNIT 17.6 - NUMERICAL MATHEMATICS 6 - NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (A)
17.6.1 Euler's unmodified method
17.6.2 Euler's modified method
17.6.3 Exercises
17.6.4 Answers to exercises (6 pages)

UNIT 17.7 - NUMERICAL MATHEMATICS 7 - NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (B)
17.7.1 Picard's method
17.7.2 Exercises
17.7.3 Answers to exercises (6 pages)

UNIT 17.8 - NUMERICAL MATHEMATICS 8 - NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (C)
17.8.1 Runge's method
17.8.2 Exercises
17.8.3 Answers to exercises (5 pages)

UNIT 18.1 - STATISTICS 1 - THE PRESENTATION OF DATA
18.1.1 Introduction
18.1.2 The tabulation of data
18.1.3 The graphical representation of data
18.1.4 Exercises
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)

Page last changed: 9 September 2002
Contact for this page: C J Judd