- 1. Basic algebra
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Mathematical Notation and Symbols
Indices
Simplification and factorisation
Arithmetic of Algebraic Fractions
Formulae and Transposition
- 2. Basic functions
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Basic Concepts of Functions
Graphs of Functions and Parametric Form
One-to-one and Inverse Functions
Characterising Functions
The Straight Line
The Circle
Some Common Functions
- 3. Equations, inequalities and partial fractions
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Solving Linear Equations
Solving Quadratic Equations
Solving Polynomial Equations
Solving Simultaneous Linear Equations
Solving Inequalities
Partial Fractions
- 4. Trigonometry
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Right-angled Triangles
Trigonometrical Functions
Trigonometrical Identities
Applications of Trigonometry to Triangles
Applications of Trigonometry to Waves
- 5. Functions and modelling
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The modelling Cycle and Functions
Quadratic Functions and Modelling
Oscillating Functions and Modelling
Inverse Square Law Modelling
- 6. Exponential and logarithmic functions
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The Exponential Function
The Hyperbolic Function
Logarithms
The Logarithmic Function
Modelling Exercises
Log-linear Graphs
- 7. Matrices
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Introduction to Matrices
Matric Multiplication
Determinants
The Inverse of a Matrix
- 8. Matric solutions of equations
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Solution by Cramer's Law
Solution by Inverse Matrix Method
Solution by Gauss Elimination
- 9. Vectors
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Basic Concepts of Vectors
Cartesian Components of Vectors
The Scalar Product
The Vector Product
Lines and Planes
- 10. Complex Numbers
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Complex Arithmetic
Argand Diagrams and the Polar Form
The Exponential Form of a Complex Number
De Moivre's Theorem
- 11. Differentiation
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Introducing Differentiation
Using a Table of Derivatives
Higher Derivatives
Differentiating Products and Quotients
The Chain Rule
Parametric Differentiation
Implicit Differentiation
- 12. Applications of Differentiation
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Tangents and Normals
Maxima and Minima
The Newton-Raphson Method
Curvature
Differentiation of Vectors
Case Study: Complex Impedance
- 13. Integration
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Basic Concepts of Integration
Definite Integrals
The area Bounded by a Curve
Integration by Parts
Integration by Substitution and using Partial Fractions
Integration of Trigonometrical Functions
- 14. Applications of Integration 1
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Integration as the Limits of a Sum
The Mean Value and Root-Mean-Square Value
Volumes of Revolution
Lengths of Curves and Surfaces of Revolution
- 15. Applications of Integration 2
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Integration of Vectors
Calculating Centres of Mass
Moments of Inertia
- 16. Sequences and Series
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Sequences and Series
Infinite Series
The Binomial Series
Power Series
Maclaurin and Taylor Series
- 17. Conics and Polar Coordinates
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Conic Sections
Polar Coordinates
Parametric Curves
- 18. Functions of Several Variables
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Functions of Several Variables
Partial Derivatives
Stationary Points
Errors and Percentage Change
- 19. Differential Equations
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Modelling with Differential Equations
First-Order Differential Equations
Second-Order Differential Equations
Applications of Differential Equations
- 20. Laplace Transforms
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Causal Functions
The Transform and its Inverse
Further Laplace Transforms
Solving Differential Equations
The Convolution Theorem
Transfer Functions
- 21. z-Transforms
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The z-Transform
Basics of z-Transform Theory
z-Transforms and Difference Equations
Engineering Applications of z-Transforms
Sampled Functions
- 22. Eigenvalues and Eigenvectors
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Basic Concepts
Applications of Eigenvalues and Eigenvectors
Repeated Eigenvalues and Symmetric Matrices
Numerical Determination of Eigenvalues and Eigenvectors
- 23. Fourier Series
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Periodic Functions
Representing Periodic Functions by Fourier Series
Even and Odd Functions
Convergence
Half-range Series
The Complex Form
An Application of Fourier Series
- 24. Fourier Transforms
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The Fourier Transform
Properties of the Fourier Transform
Some Special Fourier Transform Pairs
- 25. Partial Differential Equations
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Partial Differential Equations
Applications of PDEs
Solution using Separation of Variables
Solution using Fourier Series
- 26. Functions of a Complex Variable
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Complex Functions
Cauchy-Riemann Equations and Conformal Mapping
Standard Complex Functions
Basic Complex Integration
Cauchy's Theorem
Singularities and Residues
- 27. Multiple Integration
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Integration to Surface Integrals
Multiple Integrals over Non-rectangular Regions
Volume Integrals
Changing Coordinates
- 28. Differential Vector Calculus
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Background to Vector Calculus
Differential Vector Calculus
Orthogonal Curvilinear Coorindates
- 29. Integral Vector Calculus
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Line Integrals
Surface and Volume Integrals
Integral Vector Theorems
- 30. Introduction to Numerical Methods
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Rounding Error and Conditioning
Gaussian Elimination
LU Decomposition
Matric Norms
Iterative Methods for Systems of Equations
- 31.Numerical Methods of Approximation
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Polynomial Approximations
Numerical Integration
Numerical Differentiation
Nonlinear Equations
- 32. Numerical Initial Value Problems
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Initial Value Problems
Linear Multistep Methods
Predictor-Corrector Methods
Parabolic PDEs
Hyperbolic PDEs
- 33. Numerical Boundary Problems
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Two-point Boundary Value Problems
Elliptic PDEs
- 34. Modelling Motion
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Projectiles
Forces in More than One Dimension
Resisted Motion
- 35. Sets and Probability
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Sets
Elementary Probability
Addition and Multiplication Laws of Probability
Total Probability and Bayes Theorem
- 36. Descriptive Statistics
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Describing Data
Exploring Data
- 37. Discrete Probability Distributions
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Discrete Probability Distributions
The Binomial Distribution
The Poisson Distribution
The Hypergeometric Distribution
- 38. Continuous Probability Distributions
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Continuous Probability Distributions
The Uniform Distribution
The Exponential Distribution
- 39. The Normal Distribution
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The Normal Distribution
The Normal Approximation to the Binomial Distribution
Sums and Differences of Random Variables
- 40. Sampling Distributions and Estimation
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Sampling Distributions
Interval Estimation for the Variance
- 41. Hypothesis Testing
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Statistical Testing
Tests Concerning a Single Sample
Tests Concerning Two Samples
- 42. Goodness of Fit and Contingency Tables
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Goodness of Fit
Contingency Tables
- 43. Regression and Correlation
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Regression
Correlation
- 44. Analysis of Variance
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One-way Analysis of Variance
Two-way Analysis of Variance
Experimental Design
- 45. Non-parametric Statistics
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Non-parametric Tests for a Single Sample
Non-parametric Tests for Two Samples
- 46. Reliability and Quality Control
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Reliability
Quality Control
- 47. Mathematics and Physics Miscellany
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Dimensional Analysis in Engineering
Mathematical Explorations
Physics Case Studies
- 48. Engineering Case Studies
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Engineering Case Studies
- 49. Student's Guide
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Student's Guide
- 50. Tutor's Guide
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Tutor's Guide
