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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$67.50
