$56.25

# Further Pure Mathematics

By Mark Gaulter, Brian Gaulter, Robert Smedley, Ian Cook, Graham Upton, Thorning, Sadler

US$ 56.25

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Book Description

Following on from Introducing Pure Mathematics by Smedley and Wiseman, Further Pure Mathematics covers in one volume all the pure mathematics required by students taking further mathematics. It also provides the basics for mathematics encountered in Higher Education.

Table of Contents

- 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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