HELM
Helping Engineers Learn Mathematics
The original resource was created by the the HELM consortium and released under a creative commons licence. This derivative work is released under the same licence and is a conversion of the original into an accessible HTML format.
This conversion is a work in progress, supported by the sigma Network and undertaken by MASH at the University of Bath.
This conversion represents a best effort and is provided to the community without warranty.
For information or feedback please contact: mash-sigma@bath.ac.uk .
1. Basic Algebra
01.1 Mathematical Notation and Symbols01.2 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 Functions02.2 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 Equations03.2 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 Triangles04.2 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 Functions05.2 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 Function06.2 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 Matrices07.2 Matrix Multiplication
07.3 Determinants
07.4 The Inverse of a Matrix
8. Matrix solution of equations
08.1 Solution by Cramer’s Rule08.2 Solution by Inverse Matrix Method
08.3 Solution by Gauss Elimination
10. Complex Numbers
10.2 Argand Diagrams and the Polar Form10.3 The Exponential Form of a Complex Number
10.4 De Moivre’s Theorem
11. Differentiation
11.1 Introducing Differentiation11.2 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. Applications Of Differentiation
12.1 Tangents and Normals12.2 Maxima and Minima
12.3 The Newton-Raphson Method
12.4 Curvature
12.5 Differentiation of Vectors
12.6 Case Study: Complex Impedance
13. Integration
13.1 Basic Concepts of Integration13.2 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. Applications of integration 1
14.1 Integration as the Limit of a Sum14.2 The Mean Value and the Root-Mean-Square Value
14.3 Volumes of Revolution
14.4 Lengths of Curves and Surfaces of Revolution
15. Applications of integration 2
15.1 Integration of Vectors15.2 Calculating Centres of Mass
15.3 Moment of Inertia
16. Sequences And Series
16.1 Sequences and Series16.2 Infinite Series
16.3 The Binomial Series
16.4 Power Series
16.5 Maclaurin and Taylor Series
18. Functions Of Several Variables
18.1 Functions of Several Variables18.2 Partial Derivatives
18.3 Stationary Points
18.4 Errors and Percentage Change
19. Differential Equations
19.1 Modelling with Differential Equations19.2 First Order Differential Equations
19.3 Second Order Differential Equations
19.4 Applications of Differential Equations
22. Eigenvalues and eigenvectors
22.1 Eigenvalues and Eigenvectors22.2 Applications of Eigenvalues and Eigenvectors
22.3 Repeated Eigenvalues and Symmetric Matrices
22.4 Numerical Determination of Eigenvalues and Eigenvectors
25. Partial Differential Equations
25.1 Partial Differential Equations25.2 Applications of PDEs
25.3 Solution Using Separation of Variables
25.4 Solution Using Fourier Series
28. Differential Vector Calculus
28.1 Background to Vector Calculus28.2 Differential Vector Calculus
28.3 Orthogonal Curvilinear Coordinates
29. Integral Vector Calculus
29.1 Line Integrals Involving Vectors29.2 Surface and Volume Integrals
29.3 Integral Vector Theorems
30. Introduction to Numerical Methods
30.1 Rounding Error and Conditioning30.2 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 Approximations31.2 Numerical Integration
31.3 Numerical Differentiation
31.4 Nonlinear Equations
32. Numerical Initial Value Problems
32.1 Initial Value Problems32.2 Linear Multistep Methods
32.3 Predictor-Corrector Methods
32.4 Parabolic PDEs
32.5 Hyperbolic PDEs
37. Discrete Probability Distributions
37.1 Discrete Probability Distributions37.2 The Binomial Distribution
37.3 The Poisson Distribution
37.4 The Hypergeometric Distribution
38. Continuous Probability Distributions
38.1 Continuous Probability Distributions38.2 The Uniform Distribution
38.3 The Exponential Distribution
40. Sampling Distributions and Estimation
40.1 Sampling Distributions and Estimation40.2 Interval Estimation for the Variance
41. Hypothesis Testing
41.1 Statistical Testing41.2 Tests Concerning a Single Sample
41.3 Tests Concerning Two Samples
44. Analysis of Variance
44.1 One-Way Analysis of Variance44.2 Two-Way Analysis of Variance
44.3 Experimental Design
45. Non-parametric Statistics
45.1 Non-parametric Tests for a Single Sample45.2 Non-parametric Tests for Two Samples