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## 1. Basic Algebra

01.1 Mathematical Notation and Symbols

01.2 Indices

01.3 Simplification and Factorisation

01.4 Arithmetic of Algebraic Fractions

01.5 Formulae and Transposition

## 2. Basic Functions

02.1 Basic Concepts of Functions

02.2 Graphs of Functions and Parametric Form

02.3 One-to-One and Inverse Functions

02.4 Characterising Functions

02.5 The Straight Line

02.6 The Circle

02.7 Some Common Functions

## 3. Equations, Inequalities And Partial Fractions

03.1 Solving Linear Equations

03.2 Solving Quadratic Equations

03.3 Solving Polynomial Equations

03.4 Solving Simultaneous Linear Equations

03.5 Solving Inequalities

03.6 Partial Fractions

## 4. Trigonometry

04.1 Right-angled Triangles

04.2 Trigonometric Functions

04.3 Trigonometric Identities

04.4 Applications of Trigonometry to Triangles

04.5 Applications of Trigonometry to Waves

## 5. Functions and modelling

05.1 The Modelling Cycle and Functions

05.2 Quadratic Functions and Modelling

05.3 Oscillating Functions and Modelling

05.4 Inverse Square Law Modelling

## 6. Exponential and Logarithmic functions

06.1 The Exponential Function

06.2 The Hyperbolic Functions

06.3 Logarithms

06.4 The Logarithmic Function

06.5 Modelling Exercises

06.6 Log-linear Graphs

## 7. Matrices

07.1 Introduction to Matrices

07.2 Matrix Multiplication

07.3 Determinants

07.4 The Inverse of a Matrix

## 8. Matrix Solution of Equations

08.1 Solution by Cramer’s Rule

08.2 Solution by Inverse Matrix Method

08.3 Solution by Gauss Elimination

## 9. Vectors

09.1 Basic Concepts of Vectors

09.2 Cartesian Components of Vectors

09.3 The Scalar Product

09.4 The Vector Product

09.5 Lines and Planes

## 10. Complex Numbers

10.2 Argand Diagrams and the Polar Form

10.3 The Exponential Form of a Complex Number

10.4 De Moivre’s Theorem

## 11.Differentiation

11.1 Introducing Differentiation

11.2 Using a Table of Derivatives

11.3 Higher Derivatives

11.4 Differentiating Products and Quotients

11.5 The Chain Rule

11.6 Parametric Differentiation

11.7 Implicit Differentiation

## 12. Differentiation Applications

12.1 Tangents and Normals

12.2 Maxima and Minima

12.3 The Newton-Raphson Method

12.4 Curvature

12.5 Differentiation of Vectors

12.6 Case Study: Complex Impedance

## 13. Integrations

13.1 Basic Concepts of Integration

13.2 Definite Integrals

13.3 The Area Bounded by a Curve

13.4 Integration by Parts

13.5 Integration by Substitution and Using Partial Fractions

13.6 Integration of Trigonometric Functions

## 14. Integration Applications (1)

14.1 Integration as the Limit of a Sum

14.2 The Mean Value and the Root-Mean-Square Value

14.3 Volumes of Revolution

14.4 Lengths of Curves and Surfaces of Revolution

## 15. Integration Applications (2)

15.1 Integration of Vectors

15.2 Calculating Centres of Mass

15.3 Moment of Inertia

## 16. Sequences and Series

16.1 Sequences and Series

16.2 Infinite Series

16.3 The Binomial Series

16.4 Power Series

16.5 Maclaurin and Taylor Series

## 17. Conics and polar coordinates

17.1 Conic Sections

17.2 Polar Coordinates

17.3 Parametric Curves

## 18. Functions Of Several Variables

18.1 Functions of Several Variables

18.2 Partial Derivatives

18.3 Stationary Points

18.4 Errors and Percentage Change

## 19. Differential Equations

19.1 Modelling with Differential Equations

19.2 First Order Differential Equations

19.3 Second Order Differential Equations

19.4 Applications of Differential Equations

## 20. The Laplace Transform

20.1 Causal Functions

20.2 The Transform and its Inverse

20.3 Further Laplace Transforms

20.4 Solving Differential Equations

20.5 The Convolution Theorem

20.6 Transfer Functions

## 21. The Z transform

21.1 The z-Transform

21.2 Basics of z-Transform Theory

21.3 z-Transforms and Difference Equations

21.4 Engineering Applications of z-Transforms

21.5 Sampled Functions

## 22. Eigenvalues and Eigenvectors

22.1 Eigenvalues and Eigenvectors

22.2 Applications of Eigenvalues and Eigenvectors

22.3 Repeated Eigenvalues and Symmetric Matrices

22.4 Numerical Determination of Eigenvalues and Eigenvectors

## 23. Fourier series

23.1 Periodic Functions

23.2 Representing Periodic Functions by Fourier Series

23.3 Even and Odd Functions

23.4 Convergence

23.5 Half-Range Series

23.6 The Complex Form

23.7 An Application of Fourier Series

## 24. Fourier transforms

24.1 The Fourier transform

24.2 Properties of the Fourier Transform

24.3 Some Special Fourier transform Pairs

## 25. Partial Differential Equations

25.1 Partial Differential Equations

25.2 Applications of PDEs

25.3 Solution Using Separation of Variables

25.4 Solution Using Fourier Series

## 26. Functions of a Complex Variable

26.1 Complex Functions

26.2 Cauchy-Riemann Equations and Conformal Mapping

26.3 Standard Complex Functions

26.4 Basic Complex Integration

26.5 Cauchy’s Theorem

26.6 Singularities and Residues

## 27. Multiple Integration

27.1 Introduction to Surface Integrals

27.2 Multiple Integrals over Non-rectangular Regions

27.3 Volume Integrals

27.4 Changing Coordinates

## 28. Differential Vector Calculus

28.1 Background to Vector Calculus

28.2 Differential Vector Calculus

28.3 Orthogonal Curvilinear Coordinates

## 29. Integral Vector Calculus

29.1 Line Integrals Involving Vectors

29.2 Surface and Volume Integrals

29.3 Integral Vector Theorems

## 30. Introduction to Numerical Methods

30.1 Rounding Error and Conditioning

30.2 Gaussian Elimination

30.3 LU Decomposition

30.4 Matrix Norms

30.5 Iterative Methods for Systems of Equations

## 31. Numerical Methods of Approximation

31.1 Polynomial Approximations

31.2 Numerical Integration

31.3 Numerical Differentiation

31.4 Nonlinear Equations

## 32. Numerical Initial Value Problems

32.1 Initial Value Problems

32.2 Linear Multistep Methods

32.3 Predictor-Corrector Methods

32.4 Parabolic PDEs

32.5 Hyperbolic PDEs

## 33. Numerical Boundary Value Problems

## 34. Modelling Motion

34.1 Projectiles

34.2 Forces in More Than One Dimension

34.3 Resisted Motion

## 35. Sets and Probability

35.1 Sets

35.2 Elementary Probability

35.3 Addition and Multiplication Laws of Probability

35.4 Total Probability and Bayes’ Theorem

## 36. Descriptive Statistics

## 37. Discrete Probability Distributions

37.1 Discrete Probability Distributions

37.2 The Binomial Distribution

37.3 The Poisson Distribution

37.4 The Hypergeometric Distribution

## 38. Continuous Probability Distributions

38.1 Continuous Probability Distributions

38.2 The Uniform Distribution

38.3 The Exponential Distribution

## 39. The Normal Distribution

39.1 The Normal Distribution

39.2 The Normal Approximation to the Binomial Distribution

39.3 Sums and Differences of Random Variables

## 40. Sampling Distributions and Estimation

40.1 Sampling Distributions and Estimation

40.2 Interval Estimation for the Variance

## 41. Hypothesis Testing

41.1 Statistical Testing

41.2 Tests Concerning a Single Sample

41.3 Tests Concerning Two Samples

## 42. Goodness of Fit and Contingency Tables

## 43. Regression and Correlation

## 44. Analysis of Variance

44.1 One-Way Analysis of Variance

44.2 Two-Way Analysis of Variance

44.3 Experimental Design

## 45. Non-parametric Statistics

45.1 Non-parametric Tests for a Single Sample

45.2 Non-parametric Tests for Two Samples