Further Maths Contents
Further maths tutorials are listed here to help with your course of study
Inequalities
Fractional Inequalities
Roots of Polynomial Equations
Series
Standard Summations
Method of Differences
Maclaurin’s Series
Taylor’s Series
Infinite sequences
Infinite series
Tests for convergence
Rational Expressions
Graphs of Rational Functions
Conics
Parabola
Hyperbola (Rectangular)
The Ellipse
Matrix Algebra
Matrices
- Introduction and dimension of a matrix
- Addition and subtraction and multiplying a matrix by a scalar
- Matrix multiplication
- Identity and Inverse of a 2x2 matrix
- Exam Questions - Identity and inverse of a 2x2 matrix
- Transposed and symmetric matrices
- Finding the determinant of a 3x3 matrix
- Finding the inverse of a 3x3 matrix where it exists
Simultaneous Equations by Matrix Methods
Matrix Linear Transformations
- Linear transformations - rotations
- Linear transformations - reflections
- Linear transformations - enlargement
- How well do you know your transformations?
- Combinations of transformations
- Inverse matrices to reverse linear transformations
- Determinant as the area scale factor of a transformation
- Exam Questions - Matrix transformations
- Invariant points
Further matrix algebra
- Transposed and symmetric matrices
- Finding the determinant of a 3x3 matrix
- Linear transformations in 3 dimensions
- Using inverse matrices to reverse the effects of a linear transformation
- Eigenvalues and eigenvectors
- Reducing a symmetrical matrix to diagonal form
- Finding the inverse of a 3x3 matrix where it exists
Proof by mathematical induction
Sum of Series
Complex Numbers
Complex Numbers
- Real and imaginary numbers
- Addition, subtraction and multiplying complex numbers and simplifying powers of i
- Complex conjugates
- Division of a complex number by a complex number
- Argand diagrams
- Modulus and argument of a complex number
- Solving problems with complex numbers
- Square roots of a complex number
- Solving quadratic equations with complex roots
- Solving cubic equations
- Solving quartic equations
- Exam Questions - Complex numbers
- Geometrical effects of conjugating a complex number
- Exam Questions - Finding roots
- Modulus-argument form of a complex number
- Exponential Form (Euler's relation)
- Multiplication and division rules for mod and argument of two complex numbers
- Exam Questions - Further complex numbers
De Moivre’s Theorem
Loci in the Complex Plane
Transformations of the Complex Plane
Vectors
Triple Scalar Product
Planes
- Parametric vector form of a plane
- Scalar product forms of a plane
- Cartesian form of a plane
- Finding the point of intersection between a line and a plane
- The equation of the line of intersection between two non parallel planes
- Shortest distance from a point to a plane
- Angle between a line and a plane
- The angle between two planes
- Intersection of three planes
Systems of Linear Equations
Solutions of systems of linear equations
First Order Linear Differential Equations
Exact Equations (Integrating Factors)
Numerical Solution of Linear Differential Equations
Solving differential equations of the form dy/dx = f(x)
Second Order Linear Differential Equations
Equations of the form a d²y/dx² + b dy/dx + cy = 0
Equations of the form a d²y/dx² + b dy/dx + cy = f(x)
- General solutions where f(x) = k (constant types)
- General solutions where f(x) = kx (linear types)
- General solutions where f(x) = kx2 (quadratic types)
- General solutions where f(x) = kepx (exponential types)
- General solutions where f(x) = λ cosωx + µ sinωx (trig types)
- Special types of particular integrals
- Exam Questions - General solutions where f(x) = kx (linear types)
- Particular solutions using boundary conditions to solve differential equations
- Exam Questions - Exponential Type kepx (exponential types)
- Exam Questions - Particular solutions using boundary conditions
Using a Substitution
Polar Coordinates and Curves
Polar Coordinates
Equations of Curves
Area Bounded by a Polar Curve
Hyperbolic Functions
Hyperbolic Functions
- Definitions
- Graphs of sinh(x), cosh(x) and tanh(x)
- Graphs of sech(x), cosech(x) and coth(x)
- Solving equations using inverse and exponential functions
- Hyperbolic identities
- Osborn's rule
- Inverse hyperbolic functions and their graphs
- Expressing inverse hyperbolic functions as natural logarithms
- Solving hyperbolic equations using hyperbolic identities
Differentiation
Inverse Trigonometric Functions
Hyperbolic Functions
Differentiation – Continuity
Rolle’s Theorem
Mean Value Theorem
L’Hopital’s Rule
Integration
Standard Integrals Involving Inverse Trigonometric Functions
Standard Integrals Involving Hyperbolic Functions
Deriving and using reduction formulae
Applications of Integration – Arcs
Applications of Integration – Surface area of revolution
Binary Operations
Binary Operations
Groups
Groups
- What defines a group
- Different types of groups
- Commutative groups
- Order of a group
- Subgroups
- Lagrange’s theorem
- Cyclic groups
- Generators
- Proof that all cyclic groups are Abelian
- Permutations under composition of permutations
- Cycle notation for permutations
- Result that every permutation can be written as a composition of disjoint cycles
- The order of a combination of cycles