Further Pure Mathematics

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Description

Contents

Reviews

Language

English

ISBN

9781382018166

Front Cover

Imprint

Contents

Preface

1 Complex Numbers

What is a complex number?

Calculating with complex numbers

Argand diagram

Loci in the complex plane

Cube roots of unity

2 Further Trigonometry with Calculus

General solutions of trigonometric equations

Harmonic form

Inverse trigonometric functions

3 Polar Coordinates

Position of a point

Connection between polar and cartesian coordinates

Sketching curves given in polar coordinates

Area of a sector of a curve

Equations of the tangents to a curve

4 Differential Equations

First-order equations requiring an integrating factor

Second-order differential equations

Solution of differential equations by substitution

5 Determinants

Definition of 2 x 2 and 3 x 3 determinants

Rules for the manipulation of determinants

Factorisation of determinants

Solution of three equations in three unknowns

6 Vector Geometry

Vector equation of a line

Cartesian equation of a line

Resolved part of a vector

Direction ratios

Direction cosines

Vector product

Area of a triangle

Equation of a plane

Distance of a plane from the origin

Distance of a plane from a point

Scalar triple product and its applications

7 Curve Sketching and Inequalities

Curve sketching

Sketching rational functions with a quadratic denominator

Inequalities

8 Roots of Polynomial Equations

Roots of a quadratic equation

Roots of a cubic equation

Roots of a polynomial equation of degree n

Equations with related roots

Complex roots of a polynomial equation

9 Proof, Sequences and Series

Proof by induction

Proof by contradiction

Summation of series

Convergence

Maclaurin's series

Using power series

Power series for more complicated functions

10 Hyperbolic Functions

Definitions

Graphs of cosh x, sinh x and tanh x

Standard hyperbolic identities

Differentiation of hyperbolic functions

Integration of hyperbolic functions

Inverse hyperbolic functions

Logarithmic form

Differentiation of inverse hyperbolic functions

'Double-angle' formulae

Power series

Osborn's rule

11 Conics

Generating conics

Parabola

Ellipse

Hyperbola

Polar equation of a conic

12 Further Integration

Inverse function of a function rule

Integration by parts

Integration of fractions

Reduction formulae

Arc length

Area of a surface of revolution

Improper integrals

Summation of series

13 Numerical Methods

Solution of polynomial equations

Evaluation of areas under curves

Step-by-step solution of differential equations

Taylor's series

14 Matrices

Notation

The order of a matrix

Addition and subtraction of matrices

Multiplication of matrices

Determinant of a matrix

Identity matrices and zero matrices

Inverse matrices

Transformations

Eigenvectors and eigenvalues

Diagonalisation

The characteristic equation

15 Further Complex Numbers

De Moivre's theorem

nth roots of unity

Exponential form of a complex number

Trigonometric identities

Transformations in a complex plane

16 Intrinsic Coordinates

Trigonometric functions of ψ

Radius of curvature

Finding intrinsic equations

17 Groups

Binary and unary operations

Modular arithmetic

Definition of a group

Group table

Symmetries of a regular n-sided polygon

Non-finite groups

aⁿ notation

Permutation groups

Generator of a group

Cyclic groups

Abelian groups

Order of a group

Order of an element

Subgroups

Isomorphic groups

Lagrange's theorem

Groups of order 3

Groups of order 4

Groups of order 5

Groups of order 6

Real vector spaces

Answers

Index

Back Cover

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