Loughborough University
Leicestershire, UK
LE11 3TU
+44 (0)1509 263171
Loughborough University

Mathematics Education Centre

Activities

HELM Workbook List

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