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블로그 한눈에 보기 - 목차 (Contents) 본문
수학 수업 안내
영국 교육제도와 수학 공부
수학 모음 (Maths collection)
물리 모음 (Physics collection)
A-level Maths and Further maths
Pure mathematics Year 1
1. ALGEBRAIC EXPRESSIONS
1.1 Index laws
1.2 Expanding brackets
1.3 Factorising
1.4 Negative and fractional indices
1.5 Surds
1.6 Rationalising denominators
1.7 Mixed exercise 1
1.8 Review exercise for chapter 1
2. QUADRATICS
2.1 Solving quadratic equations
2.2 Completing the square
2.3 Functions
2.4 Quadratic graphs
2.5 The discriminant
2.6 Modelling with quadratics
2.7 Mixed exercise 2
2.8 Review exercise for chapter 2
3. EQUATIONS AND INEQUALITIES
3.1 Linear simultaneous equations
3.2 Quadratic simultaneous equations
3.3 Simultaneous equations on graphs
3.4 Linear inequalities
3.5 Quadratic inequalities
3.6 Inequalities on graphs
3.7 Regions
3.8 Mixed exercise 3
3.9 Review exercise for chapter 3
4. GRAPHS AND TRANSFORMATIONS
4.1 Cubic graphs
4.2 Quartic graphs
4.3 Reciprocal graphs
4.4 Points of intersection
4.5 Translating graphs
4.6 Stretching graphs
4.7 Transforming functions
4.8 Mixed exercise 4
4.9 Review exercise for chapter 4
5. STRAIGHT LINE GRAPHS
5.1 $y=mx+c$
5.2 Equations of straight lines
5.3 Parallel and perpendicular lines
5.4 Length and area
5.5 Modelling with straight lines
5.6 Mixed exercise 5
5.7 Review exercise for chapter 5
6. CIRCLES
6.1 Mid-points and perpendicular bisectors
6.2 Equations of a circle
6.3 Intersections of straight lines and circles
6.4 Use tangent and chord properties
6.5 Circles and triangles
6.6 Mixed exercise 6
6.7 Review exercise for chapter 6
7. ALGEBRAIC METHODS
7.1 Algebraic fractions
7.2 Dividing polynomials
7.3 The factor theorem
7.4 Mathematical proof
7.5 Methods of proof
7.6 Mixed exercise 7
7.7 Review exercise for chapter 7
8. THE BINOMIAL EXPANSION
8.1 Pascal’s triangle
8.2 Factorial notation
8.3 The binomial expansion
8.4 Solving binomial problems
8.5 Binomial estimation
8.6 Mixed exercise 8
8.7 Review exercise for chapter 8
9. TRIGONOMETRIC RATIOS
9.1 The cosine rule
9.2 The sine rule
9.3 Areas of triangles
9.4 Solving triangle problems
9.5 Graphs of sine, cosine and tangent
9.6 Transforming trigonometric graphs
9.7 Mixed exercise 9
9.8 Review exercise for chapter 9
10. TRIGONOMETRIC IDENTITIES AND EQUATIONS
10.1 Angles in all four quadrants
10.2 Exact values of trigonometric ratios
10.3 Trigonometric identities
10.4 Simple trigonometric equations
10.5 Harder trigonometric equations
10.6 Equations and identities
10.7 Mixed exercise 10
10.8 Review exercise for chapter 10
11. VECTORS
11.1 Vectors
11.2 Representing vectors
11.3 Magnitude and direction
11.4 Position vectors
11.5 Solving geometric problems
11.6 Modelling with vectors
11.7 Mixed exercise 11
11.8 Review exercise for chapter 11
12. DIFFERENTIATION
12.1 Gradients of curves
12.2 Finding the derivative
12.3 Differentiating $x^n$
12.4 Differentiating quadratics
12.5 Differentiating functions with two or more terms
12.6 Gradients, tangents and normal
12.7 Increasing and decreasing functions
12.8 Second order derivatives
12.9 Stationary points
12.10 Sketching gradient functions
12.11 Modelling with differentiation
12.12 Mixed exercise 12
12.13 Review exercise for chapter 12
13. INTEGRATION
13.1 Integrating $x^n$
13.2 Indefinite integrals
13.3 Finding functions
13.4 Definite integrals
13.5 Areas under curves
13.6 Areas under the x-axis
13.7 Areas between curves and lines
13.8 Mixed exercise 13
13.9 Review exercise for chapter 13
14. EXPONENTIALS AND LOGARITHMS
14.1 Exponential functions
14.2 $y=\textrm{e}^x$
14.3 Exponential modelling
14.4 Logarithms
14.5 Laws of logarithms
14.6 Solving equations using logarithms
14.7 Working with natural logarithms
14.8 Logarithms and non-linear data
14.9 Mixed exercise 14
14.10 Review exercise for chapter 14
Pure mathematics Year 2
1. ALGEBRAIC METHODS
1.1 Proof by contradiction
1.2 Algebraic fractions
1.3 Partial fractions
1.4 Repeated factors
1.5 Algebraic division
1.6 Mixed exercise 1
1.7 Review exercise for chapter 1
2. FUNCTIONS AND GRAPHS
2.1 The modulus function
2.2 Functions and mappings
2.3 Composite functions
2.4 Inverse functions
2.5 $ y = \vert \textrm f(x) \vert $ and $ y = \textrm f(\vert x\vert) $
2.6 Combining transformations
2.7 Solving modulus problems
2.8 Mixed exercise 2
2.9 Review exercise for chapter 2
3. SEQUENCES AND SERIES
3.1 Arithmetic sequences
3.2 Arithmetic series
3.3 Geometric sequences
3.4 Geometric series
3.5 Sum to infinity
3.6 Sigma notation
3.7 Recurrence relations
3.8 Modelling with series
3.9 Mixed exercise 3
3.10 Review exercise for chapter 3
4. BINOMIAL EXPANSION
4.1 Expanding $(1+x)^n$
4.2 Expanding $(a+bx)^n$
4.3 Using partial fractions
4.4 Mixed exercise 4
4.5 Review exercise for chapter 4
5. RADIANS
5.1 Radian measure
5.2 Arc length
5.3 Areas of sectors and segments
5.4 Solving trigonometric equations
5.5 Small angle approximations
5.6 Mixed exercise 5
5.7 Review exercise for chapter 5
6. TRIGONOMETRIC FUNCTIONS
6.1 Secant, cosecant and cotangent
6.2 Graphs of $\sec x$, $\csc x$ and $\cot x$
6.3 Using $\sec x$, $\csc x$ and $\cot x$
6.4 Trigonometric identities
6.5 Inverse trigonometric functions
6.6 Mixed exercise 6
6.7 Review exercise for chapter 6
7. TRIGONOMETRY AND MODELLING
7.1 Addition formulae
7.2 Using the angle addition formulae
7.3 Double-angle formulae
7.4 Solving trigonometric equations
7.5 Simplifying $a \cos x \pm b \sin x$ (Harmonic forms/identities)
7.6 Proving trigonometric identities
7.7 Modelling with trigonometric functions
7.8 Mixed exercise 7
7.9 Review exercise for chapter 7
8. PARAMETRIC EQUATIONS
8.1 Parametric equations
8.2 Using trigonometric identities
8.3 Curve sketching
8.4 Points of interection
8.5 Modelling with parametric equations
8.6 Mixed exercise 8
8.7 Review exercise for chapter 8
9. DIFFERENTIATION
9.1 Differentiating $\sin x$ and $\cos x$
9.2 Differentiating exponentials and logarithms
9.3 The chain rule
9.4 The product rule
9.5 The quotient rule
9.6 Differentiating trigonometric functions
9.7 Parametric differentiation
9.8 Implicit differentiation
9.9 Using second derivatives
9.10 Rates of change
9.11 Mixed exercise 9
9.12 Review exercise for chapter 9
10. NUMERICAL METHODS
10.1 Locating roots
10.2 Iteration 1 - Interval bisection
10.3 Iteration 2 - Fixed-point method
10.4 Iteration 3 - The Newton-Raphson method
10.5 Applications to modelling
10.6 Mixed exercise 10
10.7 Review exercise for chapter 10
11. INTEGATION
11.1 Integrating standard functions
11.2 Reverse chain rule 1: Integrating ${\rm f}(ax+b)$
11.3 Using trigonometric identities
11.4 Reverse chain rule 2: Integrating ${\rm f}'(x)/{\rm f}(x)$ and ${\rm f}'(x)[{\rm f}(x)]^n$
11.5 Integration by substitution (Reverse chain rule)
11.6 Integration by parts (Reverse product rule)
11.7 Partial fractions
11.8 Finding areas
11.9 The trapezium rule
11.10 Solving differential equations
11.11 Modelling with differential equations
11.12 Mixed exercise 11
11.13 Review exercise for chapter 11
12. VECTORS
12.1 3D coordinates
12.2 Vectors in 3D
12.3 Solving geometric problems
12.4 Application to mechanics
12.5 Mixed exercise 12
12.6 Review exercise for chapter 12
Core pure mathematics 1 (Pure mathematics 3)
1. COMPLEX NUMBERS
1.1 Imaginary and complex numbers
1.2 Multiplying complex numbers
1.3 Complex conjugation
1.4 Roots of quadratic equations
1.5 Solving cubic and quartic equations
1.6 Mixed exercise 1
1.7 Review exercise for chapter 1
2. ARGAND DIAGRAMS
2.1 Argand diagrams
2.2 Modulus and argument
2.3 Modulus-argument form of complex numbers
2.4 Loci in the Argand diagram
2.5 Regions in the Argand diagram
2.6 Mixed exercise 2
2.7 Review exercise for chapter 2
3. SERIES
3.1 Sums of natural numbers
3.2 Sums of squares and cubes
3.3 Mixed exercise 3
3.4 Review exercise for chapter 3
4. ROOTS OF POLYNOMIALS
4.1 Roots of a quadratic equation
4.2 Roots of a cubic equation
4.3 Roots of a quartic equation
4.4 Expressions relating to the roots of a polynomial
4.5 Linear transformations of roots
4.6 Mixed exercise 4
4.7 Review exercise for chapter 4
5. VOLUMES OF REVOLUTION
5.1 Volumes of revolution around the x-axis
5.2 Volumes of revolution around the y-axis
5.3 Adding and subtracting volumes
5.4 Modelling with volumes of revolution
5.5 Mixed exercise 5
5.6 Review exercise for chapter 5
6. MATRICES
6.1 Introduction to matrices
6.2 Matrix multiplication
6.3 Determinants
6.4 Inverting a $2\times2$ matrix
6.5 Inverting a $3\times3$ matrix
6.6 Solving systems of equations using matrices
6.7 Mixed exercise 6
6.8 Review exercise for chapter 6
7. LINEAR TRANSFORMATIONS
7.1 Linear transformations in two dimensions
7.2 Reflections and rotations
7.3 Enlargements and stretches
7.4 Successive transformations
7.5 Linear transformations in three dimensions
7.6 The inverse of a linear transformation
7.7 Mixed exercise 7
7.8 Review exercise for chapter 7
8. PROOF BY INDUCTION
8.1 Proof by mathematical induction
8.2 Proving divisibility results
8.3 Proving statements involving matrices
8.4 Mixed exercise 8
8.5 Review exercise for chapter 8
9. VECTORS
9.1 Equation of a line in three dimensions
9.2 Equation of a plane in three dimensions
9.3 Scalar product
9.4 Calculating angles between lines and planes
9.5 Points of intersection
9.6 Finding perpendiculars
9.7 Mixed exercise 9
9.8 Review exercise for chapter 9
Core pure mathematics 2 (Pure mathematics 4)
1. COMPLEX NUMBERS
1.1 Exponential form of complex numbers
1.2 Multiplying and dividing complex numbers
1.3 De Moivre's theorem
1.4 Trigonometric identities
1.5 Sums of series
1.6 $n$th roots of a complex number
1.7 Solving geometric problems
1.8 Mixed exercise 1
1.9 Review exercise for chapter 1
2. SERIES
2.1 The method of differences
2.2 Higher derivatives
2.3 Maclaurin series
2.4 Series expansions of compound functions
2.5 Mixed exercise 2
2.6 Review exercise for chapter 2
3. METHODS IN CALCULUS
3.1 Improper integrals
3.2 The mean value of a function
3.3 Differentiating inverse trigonometric functions
3.4 Integrating with inverse trigonometric functions
3.5 Integrating using partial fractions
3.6 Mixed exercise 3
3.7 Review exercise for chapter 3
4. VOLUMES OF REVOLUTION
4.1 Volumes of revolution around the $x$-axis
4.2 Volumes of revolution around the $y$-axis
4.3 Volumes of revolution of parametrically defined curves
4.4 Modelling with volumes of revolution
4.5 Mixed exercise 4
4.6 Review exercise for chapter 4
5. POLAR COORDINATES
5.1 Polar coordinates and equations
5.2 Sketching curves
5.3 Areas enclosed by a polar curve
5.4 Tangents to polar curves
5.5 Mixed exercise 5
5.6 Review exercise for chapter 5
6. HYPERBOLIC FUNCTIONS
6.1 Introduction to hyperbolic functions
6.2 Inverse hyperbolic functions
6.3 Identities and equations
6.4 Differentiating hyperbolic functions
6.5 Integrating hyperbolic functions
6.6 Mixed exercise 6
6.7 Review exercise for chapter 6
7. METHODS IN DIFFERENTIAL EQUATIONS
7.1 First-order differential equations
7.2 Second-order homogeneous differential equations
7.3 Second-order non-homogeneous differential equations
7.4 Using boundary conditions
7.5 Mixed exercise 7
7.6 Review exercise for chapter 7
8. MODELLING WITH DIFFERENTIAL EQUATIONS
8.1 Modelling with first-order differential equations
8.2 Simple harmonic motion
8.3 Damped and forced harmonic motion
8.4 Coupled first-order simultaneous differential equations
8.5 Mixed exercise 8
8.6 Review exercise for chapter 8
Further pure mathematics 1 (Pure mathematics 5)
1. VECTORS
1.1 Vector product
1.2 Finding areas
1.3 Scalar triple product
1.4 Straight lines
1.5 Solving geometrical problems
1.6 Mixed exercise 1
1.7 Review exercise for chapter 1
2. CONIC SECTIONS 1
2.1 Parametric equations
2.2 Parabolas
2.3 Rectangular hyperbolas
2.4 Tangents and normals
2.5 Loci
2.6 Mixed exercise 2
2.7 Review exercise for chapter 2
3. CONIC SECTIONS 2
3.1 Ellipses
3.2 Hyperbolas
3.3 Eccentricity
3.4 Tangents and normals to an ellipse
3.5 Tangents and normals to a hyperbola
3.6 Loci
3.7 Mixed exercise 3
3.8 Review exercise for chapter 3
4. INEQUALITIES
4.1 Algebraic methods
4.2 Using graphs to solve inequalities
4.3 Modulus inequalities
4.4 Mixed exercise 4
4.5 Review exercise for chapter 4
5. THE t-FORMULAE
5.1 The $t$-formulae
5.2 Applying the $t$-formulae to trigonometric identities
5.3 Solving trigonometric equations
5.4 Modelling with trigonometry
5.5 Mixed exercise 5
5.6 Review exercise for chapter 5
6. TAYLOR SERIES
6.1 Taylor series
6.2 Finding limits
6.3 Series solutions of differential equations
6.4 Mixed exercise 6
6.5 Review exercise for chapter 6
7. METHODS IN CALCULUS
7.1 Leibnitz's theorem and nth derivatives
7.2 L'Hospital's rule
7.3 The Weierstrass substitution
7.4 Mixed exercise 7
7.5 Review exercise for chapter 7
8. NUMERICAL METHODS
8.1 Solving first-order differential equations
8.2 Solving second-order differential equations
8.3 Simpson's rule
8.4 Mixed exercise 8
8.5 Review exercise for chapter 8
9. REDUCIBLE DIFFERENTIAL EQUATIONS
9.1 First-order differential equations
9.2 Second-order differential equations
9.3 Modelling with differential equations
9.4 Mixed exercise 9
9.5 Review exercise for chapter 9
Further pure mathematics 2 (Pure mathematics 6)
1. NUMBER THEORY
1.1 The division algorithm
1.2 The Euclidean algorithm
1.3 Modular arithmetic
1.4 Divisibility tests
1.5 Solving congruence equations
1.6 Fermat's little theorem
1.7 Combinatorics
1.8 Mixed exercise 1
1.9 Review exercise for chapter 1
2. GROUPS
2.1 The axioms for a group
2.2 Cayley tables and finite groups
2.3 Order and subgroups
2.4 Isomorphism
2.5 Mixed exercise 2
2.6 Review exercise for chapter 2
3. COMPLEX NUMBERS
3.1 Loci in an Argand diagram
3.2 Regions in an Argand diagram
3.3 Transformations of the complex plane
3.4 Mixed exercise 3
3.5 Review exercise for chapter 3
4. RECURRENCE RELATIONS
4.1 Forming recurrence relations
4.2 Solving first-order recurrence relations
4.3 Solving second-order recurrence relations
4.4 Proving closed forms
4.5 Mixed exercise 4
4.6 Review exercise for chapter 4
5. MATRIX ALGEBRA
5.1 Eigenvalues and eigenvectors
5.2 Reducing matrices to diagonal form
5.3 The Cayley-Hamilton theorem
5.4 Mixed exercise 5
5.5 Review exercise for chapter 5
6. INTEGRATION TECHNIQUES
6.1 Reduction formulae
6.2 Arc length
6.3 Area of a surface of revolution
6.4 Mixed exercise 6
6.5 Review exercise for chapter 6
Mechanics Year 1
1.1 Constructing a model
1.2 Modelling assumptions
1.3 Quantities and units
1.4 Working with vectors
1.5 Mixed exercise 1
1.6 Review exercise for chapter 1
2. CONSTANT ACCELERATION
2.1 Displacement-time graphs
2.2 Velocity-time graphs
2.3 Constant acceleration formulae 1
2.4 Constant acceleration formulae 2
2.5 Vertical motion under gravity
2.6 Mixed exercise 2
2.7 Review exercise for chapter 2
3. FORCES AND MOTION
3.1 Force diagrams
3.2 Forces as vectors
3.3 Forces and acceleration
3.4 Motion in 2 dimensions
3.5 Connected particles
3.6 Pulleys
3.7 Mixed exercise 3
3.8 Review exercise for chapter 3
4. VARIABLE ACCELERATION
4.1 Functions of time
4.2 Using differentiation
4.3 Maxima and minima problems
4.4 Using integration
4.5 Constant acceleration formulae
4.6 Mixed exercise 4
4.7 Review exercise for chapter 4
Mechanics Year 2
1.1 Moments
1.2 Resultant moments
1.3 Equilibrium
1.4 Centres of mass
1.5 Tilting
1.6 Mixed exercise 1
1.7 Review exercise for chapter 1
2. FORCES AND FRICTION
2.1 Resolving forces
2.2 Inclined planes
2.3 Friction
2.4 Mixed exercise 2
2.5 Review exercise for chapter 2
3. PROJECTILES
3.1 Horizontal projection
3.2 Horizontal and vertical components
3.3 Projection at any angle
3.4 Projectile motion formulae
3.5 Mixed exercise 3
3.6 Review exercise for chapter 3
4. APPLICATIONS OF FORCES
4.1 Static particles
4.2 Modelling with statics
4.3 Friction and static particles 4.4 Static rigid particles
4.5 Dynamics and inclined planes
4.6 Connected particles
4.7 Mixed exercise 4
4.8 Review exercise for chapter 4
5. FURTHER KINEMATICS
5.1 Vectors in kinematics
5.2 Vector methods with projectiles
5.3 Variable acceleration in one dimension
5.4 Differentiating vectors
5.5 Integrating vectors
5.6 Mixed exercise 5
5.7 Review exercise for chapter 5
Further mechanics 1
1.1 Momentum in one direction
1.2 Conservation of momentum
1.3 Momentum as a vector
1.4 Mixed exercise 1
1.5 Review exercise for chapter 1
2. WORK, ENERGY AND POWER
2.1 Work done
2.2 Kinetic and potential energy
2.3 Conservation of mechanical energy and the work-energy principle
2.4 Power
2.5 Mixed exercise 2
2.6 Review exercise for chapter 2
3. ELASTIC STRINGS AND SPRINGS
3.1 Hooke's law and equilibrium problems
3.2 Hooke's law and dynamics problems
3.3 Elastic energy
3.4 Problems involving elastic energy
3.5 Mixed exercise 3
3.6 Review exercise for chapter 3
4. ELASTIC COLLISION IN ONE DIMENSION
4.1 Direct impact and Newton’s law of restitution
4.2 Direct collision with a smooth plane
4.3 Loss of kinetic energy
4.4 Successive direct impacts
4.5 Mixed exercise 4
4.6 Review exercise for chapter 4
5. ELASTIC COLLISIONS IN TWO DIMENSIONS
5.1 Oblique impact with a fixed surface
5.2 Successive oblique impacts
5.3 Oblique impact of smooth spheres
5.4 Mixed exercise 5
5.5 Review exercise for chapter 5
Further mechanics 2
1.1 Angular speed
1.2 Acceleration of an object moving on a horizontal circular path
1.3 Objects moving in vertical circles
1.4 Objects moving in vertical circles
1.5 Objects not constrained on a circular path
1.6 Mixed exercise 1
1.7 Review exercise for chapter 1
2. CENTRES OF MASS OF PLANE FIGURES
2.1 Centre of mass of a set of particles on a straight line
2.2 Centre of mass of a set of particles arranged in a plane
2.3 Centres of mass of standard uniform plane laminas
2.4 Centre of mass of a composite lamina
2.5 Centre of mass of a framework
2.6 Lamina in equilibrium
2.7 Frameworks in equilibrium
2.8 Non-uniform composite laminas and frameworks
2.9 Mixed exercise 2
2.10 Review exercise for chapter 2
3. FURTHER CENTRES OF MASS
3.1 Using calculus to find centres of mass
3.2 Centres of mass of a uniform body
3.3 Non-uniform bodies
3.4 Rigid bodies in equilibrium
3.5 Toppling and sliding
3.6 Mixed exercise 3
3.7 Review exercise for chapter 3
4. KINEMATICS
4.1 Acceleration varying wtih time
4.2 Acceleration varying with displacement
4.3 Acceleration varying with velocity
4.4 Mixed exercise 4
4.5 Review exercise for chapter 4
5. DYNAMICS
5.1 Motion in a straight line with variable force
5.2 Newton's law of gravitation
5.3 Simple harmonic motion
5.4 Horizontal oscillation
5.5 Vertical oscillation
5.6 Mixed exercise 5
5.7 Review exercise for chapter 5
Statistics Year 1
1.1 Populations and samples
1.2 Sampling
1.3 Non-random sampling
1.4 Types of data
1.5 The large data set
1.6 Mixed exercise 1
1.7 Review exercise for chapter 1
2. MEASURES OF LOCATION AND SPREAD
2.1 Measures of central tendency
2.2 Other measures of location
2.3 Measures of spread
2.4 Variance and standard deviation
2.5 Coding
2.6 Mixed exercise 2
2.7 Review exercise for chapter 2
3. REPRESENTATIONS OF DATA
3.1 Outliers
3.2 Box plots
3.3 Cumulative frequency
3.4 Histograms
3.5 Comparing data
3.6 Mixed exercise 3
3.7 Review exercise for chapter 3
4. CORRELATION
4.1 Correlation
4.2 Linear regression
4.3 Mixed exercise 4
4.4 Review exercise for chapter 4
5. PROBABILITY
5.1 Calculating probabilities
5.2 Venn diagrams
5.3 Mutually exclusive and independent events
5.4 Tree diagrams
5.5 Mixed exercise 5
5.6 Review exercise for chapter 5
6. STATISTICAL DISTRIBUTIONS
6.1 Probability distributions
6.2 The binomial distribution
6.3 Cumulative probabilities
6.4 Mixed exercise 6
6.5 Review exercise for chapter 6
7. HYPOTHESIS TESTING
7.1 Hypothesis testing
7.2 Finding critical values
7.3 One-tailed tests
7.4 Two-tailed tests
7.5 Mixed exercise 6
7.6 Review exercise for chapter 6
Statistics Year 2
1.1 Exponential models
1.2 Measuring correlation
1.3 Hypothesis testing for zero correlation
1.4 Mixed exercise 1
1.5 Review exercise for chapter 1
2. CONDITIONAL PROBABILITY
2.1 Set notation
2.2 Conditional probability
2.3 Conditional probabilities in Venn diagrams
2.4 Probability formulae
2.5 Tree diagrams
2.6 Mixed exercise 2
2.7 Review exercise for chapter 2
3. THE NORMAL DISTRIBUTION
3.1 The normal distribution
3.2 Finding probabilities for normal distributions
3.3 The inverse normal distribution function
3.4 The standard normal distribution
3.5 Finding $\mu$ and $\sigma$
3.6 Approximating a binomial distribution
3.7 Hypothesis testing with the normal distribution
3.8 Mixed exercise 3
3.9 Review exercise for chapter 3
Further statistics 1
1.1 Expected value of a discrete random variable
1.2 Variance of a discrete random variable
1.3 Expected value and variance of a function of $X$
1.4 Solving problems involving random variables
1.5 Mixed exercise 1
1.6 Review exercise for chapter 1
2. POISSON DISTRIBUTIONS
2.1 The Poisson distribution
2.2 Modelling with the Poisson distribution
2.3 Adding Poisson distributions
2.4 Mean and variance of a Poisson distribution
2.5 Mean and variance of the binomial distribution
2.6 Using the Poisson distribution to approximate the binomial distribution
2.7 Mixed exercise 2
2.8 Review exercise for chapter 2
3. GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS
3.1 The geometric distribution
3.2 Mean and variance of a geometric distribution
3.3 The negative binomial distribution
3.4 Mean and variance of the negative binomial distribution
3.5 Mixed exercise 3
3.6 Review exercise for chapter 3
4. HYPOTHESIS TESTING
4.1 Testing for the mean of a Poisson distribution
4.2 Finding critical regions for a Poisson distribution
4.3 Hypothesis testing for the parameter p of a geometric distribution
4.4 Finding critical regions for a geometric distribution
4.4 Mixed exercise 4
4.5 Review exercise for chapter 4
5. CENTRAL LIMIT THEOREM
5.1 The central limit theorem
5.2 Applying the central limit theorem to other distributions
5.3 Mixed exercise 5
5.4 Review exercise for chapter 5
6. CHI-SQUARED TESTS
6.1 Goodness of fit
6.2 Degrees of freedom and the chi-squared family of distributions
6.3 Testing a hypothesis
6.4 Testing the goodness of fit with discrete data
6.5 Using contingency tables
6.6 Applying goodness-of-fit tests to geometric distributions
6.7 Mixed exercise 6
6.8 Review exercise for chapter 6
7. PROBABILITY GENERATING FUNCTIONS
7.1 Probability generating functions
7.2 Probability generating functions of standard distributions
7.3 Mean and variance of a distribution
7.4 Sums of independent random variables
7.5 Mixed exercise 7
7.6 Review exercise for chapter 7
8. QUALITY OF TESTS
8.1 Type I and Type II errors
8.2 Finding Type I and Type II errors using the normal distribution
8.3 Calculate the size and power of a test
8.4 The power function
8.5 Mixed exercise 8
8.6 Review exercise for chapter 8
Further statistics 2
1.1 Least squares linear regression
1.2 Residuals
1.3 Mixed exercise 1
1.4 Review exercise for chapter 1
2. CORRELATION
2.1 The product moment correlation coefficient
2.2 Spearman’s rank correlation coefficient
2.3 Hypothesis testing for zero correlation
2.4 Mixed exercise 2
2.5 Review exercise for chapter 2
3. CONTINUOUS DISTRIBUTIONS
3.1 Continuous random variables
3.2 The cumulative distribution function
3.3 Mean and variance of a continuous distribution
3.4 Mode, median, percentiles and skewness
3.5 The continuous uniform distribution
3.6 Modelling with the continuous uniform distribution
3.7 Mixed exercise 3
3.8 Review exercise for chapter 3
4. COMBINATIONS OF RANDOM VARIABLES
4.1 Combinations of random variables
4.2 Mixed exercise 4
4.3 Review exercise for chapter 4
5. ESTIMATION, CONFIDENCE INTERVALS AND TESTS USING A NORMAL DISTRIBUTION
5.1 Estimators, bias and standard error
5.2 Confidence intervals
5.3 Hypothesis testing for the difference between means
5.4 Use of large sample results for an unknown population
5.5 Mixed exercise 5
5.6 Review exercise for chapter 5
6. FURTHER HYPOTHESIS TESTS
6.1 Variance of a normal distribution
6.2 Hypothesis testing for the variance of a normal distribution
6.3 The $F$-distribution
6.4 The $F$-test
6.5 Mixed exercise 6
6.6 Review exercise for chapter 6
7. CONFIDENCE INTERVALS AND TESTS USING THE $t$-DISTRIBUTION
7.1 Mean of a normal distribution with unknown variance
7.2 Hypothesis test for the mean of a normal distribution with unknown variance
7.3 The paired $t$-test
7.4 Difference between means of two independent normal distributions
7.5 Hypothesis test for the difference between means
7.6 Mixed exercise 7
7.7 Review exercise for chapter 7
University Maths & Theoretical Physics
(Under construction...)
First-year courses
IA Vectors and matrices
IA Group theory
IA Numbers and sets
IA Differential equations
IA Vector calculus
IA Probability
IA Dynamics
IA Analysis
Second-year courses
IB Linear algebra
IB Analysis II
IB Groups, rings and modules
IB Geometry
IB Mathematical methods
IB Complex methods
IB Electromagnetism
IB Quantum mechanics
IB Fluid dynamics
IB Variational principles
IB Statistics
IB Markov chains
IB Numerical analysis
IB Optimisation
Third-year courses
II Further complex methods
II Asymptotic methods
II Classical dynamics
II Dynamical systems
II Electrodynamics
II Statistical physics
II Principles of quantum mechanics
II Applications of quantum mechanics
II Quantum information and computation
II Integrable systems
II General relativity
II Cosmology
II Fluid dynamics
II Waves
II Automata and formal languages
II Mathematics of machine learning
II Number theory
II Topics in analysis
II Coding and cryptography
II Statistical modelling
II Mathematical biology
II Logic and set theory
II Graph theory
II Galois theory
II Representation theory
II Number fields
II Algebraic topology
II Linear analysis
II Analysis of functions
II Riemann surfaces
II Algebraic geometry
II Differential geometry
II Probability and measure
II Applied probability
II Principles of statistics
II Stochastic financial models
II Numerical analysis
Fourth-year courses
Particle Physics and Quantum Fields
III Quantum Field Theory
III Symmetries, Particles and Fields
III Statistical Field Theory
III Non-equilibrium Statistical Field Theory
III Advanced Quantum Field Theory
III Standard Model
III String Theory
III Supersymmetry
III Physics beyond the Standard Model
Relativity and Cosmology
III General Relativity
III Black Holes
III Cosmology
III Advanced Cosmology
III Field Theory in Cosmology
III Solitons, Instantons and Geometry
III Applications of Differential Geometry in Physics
III Applications of Analysis in Physics
III Gauge-gravity duality
Astrophysics
III Structure and Evolution of Stars
III Astrostatistics
III Extrasolar Planets: Atmospheres and Interiors
III Astrophysical Black Holes
III Binary stars
III Dynamics of Astrophysical Discs
III Modern Stellar Dynamics
Quantum Computation, Information and Foundations
III Quantum Information Theory
III Quantum Computation
III Quantum Information Foundations and Gravity
Continuum Mechanics
III Biological Physics and Fluid Dynamics
III Fluid Dynamics of Climate
III Slow Viscous Flow
III Theoretical Physics of Soft Condensed Matter
III Fluid Dynamics of the Environment
III Fluid Dynamics of the Solid Earth
III Perturbation Methods
III Stochastic Processes in Theoretical Physics and Biology
Applied and Computational Analysis
III Unbounded Operators and Semigroups
III Numerical Solution of Differential Equations
III Inverse Problems
III Topics in Convex Optimisation
III Mathematical Analysis of the Incompressible Navier-Stokes Equations
III Introduction to Nonlinear Spectral Analysis