# Help Engineers Learn Mathematics (HELM) workbooks

HELM (Helping Engineers Learn Mathematics) Workbooks have been developed to assist engineering students to learn the mathematics and statistics that they will need at university.

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1. Basic algebra

Mathematical Notation and Symbols

Indices

Simplification and factorisation

Arithmetic of Algebraic Fractions

Formulae and Transposition

2. Basic functions

Basic Concepts of Functions

Graphs of Functions and Parametric Form

One-to-one and Inverse Functions

Characterising Functions

The Straight Line

Some Common Functions

3. Equations, inequalities and partial fractions

Solving Linear Equations

Solving Polynomial Equations

Solving Simultaneous Linear Equations

Solving Inequalities

Partial Fractions

4. Trigonometry

Right-angled Triangles

Trigonometrical Functions

Trigonometrical Identities

Applications of Trigonometry to Triangles

Applications of Trigonometry to Waves

5. Functions and modelling

The modelling Cycle and Functions

Oscillating Functions and Modelling

Inverse Square Law Modelling

6. Exponential and logarithmic functions

The Exponential Function

The Hyperbolic Function

Logarithms

The Logarithmic Function

Modelling Exercises

Log-linear Graphs

7. Matrices

Introduction to Matrices

Matric Multiplication

Determinants

The Inverse of a Matrix

8. Matric solutions of equations

Solution by Cramer's Law

Solution by Inverse Matrix Method

Solution by Gauss Elimination

9. Vectors

Basic Concepts of Vectors

Cartesian Components of Vectors

The Scalar Product

The Vector Product

Lines and Planes

10. Complex Numbers

Complex Arithmetic

Argand Diagrams and the Polar Form

The Exponential Form of a Complex Number

De Moivre's Theorem

11. Differentiation

Introducing Differentiation

Using a Table of Derivatives

Higher Derivatives

Differentiating Products and Quotients

The Chain Rule

Parametric Differentiation

Implicit Differentiation

12. Applications of Differentiation

Tangents and Normals

Maxima and Minima

The Newton-Raphson Method

Curvature

Differentiation of Vectors

Case Study: Complex Impedance

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 Trigonometrical Functions

14. Applications of Integration 1

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

Integration of Vectors

Calculating Centres of Mass

Moments of Inertia

16. Sequences and Series

Sequences and Series

Infinite Series

The Binomial Series

Power Series

Maclaurin and Taylor Series

17. Conics and Polar Coordinates

Conic Sections

Polar Coordinates

Parametric Curves

18. Functions of Several Variables

Functions of Several Variables

Partial Derivatives

Stationary Points

Errors and Percentage Change

19. Differential Equations

Modelling with Differential Equations

First-Order Differential Equations

Second-Order Differential Equations

Applications of Differential Equations

20. Laplace Transforms

Causal Functions

The Transform and its Inverse

Further Laplace Transforms

Solving Differential Equations

The Convolution Theorem

Transfer Functions

21. z-Transforms

The z-Transform

Basics of z-Transform Theory

z-Transforms and Difference Equations

Engineering Applications of z-Transforms

Sampled Functions

22. Eigenvalues and Eigenvectors

Basic Concepts

Applications of Eigenvalues and Eigenvectors

Repeated Eigenvalues and Symmetric Matrices

Numerical Determination of Eigenvalues and Eigenvectors

23. Fourier Series

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

The Fourier Transform

Properties of the Fourier Transform

Some Special Fourier Transform Pairs

25. Partial Differential Equations

Partial Differential Equations

Applications of PDEs

Solution using Separation of Variables

Solution using Fourier Series

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

27. Multiple Integration

Integration to Surface Integrals

Multiple Integrals over Non-rectangular Regions

Volume Integrals

Changing Coordinates

28. Differential Vector Calculus

Background to Vector Calculus

Differential Vector Calculus

Orthogonal Curvilinear Coorindates

29. Integral Vector Calculus

Line Integrals

Surface and Volume Integrals

Integral Vector Theorems

30. Introduction to Numerical Methods

Rounding Error and Conditioning

Gaussian Elimination

LU Decomposition

Matric Norms

Iterative Methods for Systems of Equations

31.Numerical Methods of Approximation

Polynomial Approximations

Numerical Integration

Numerical Differentiation

Nonlinear Equations

32. Numerical Initial Value Problems

Initial Value Problems

Linear Multistep Methods

Predictor-Corrector Methods

Parabolic PDEs

Hyperbolic PDEs

33. Numerical Boundary Problems

Two-point Boundary Value Problems

Elliptic PDEs

34. Modelling Motion

Projectiles

Forces in More than One Dimension

Resisted Motion

35. Sets and Probability

Sets

Elementary Probability

Addition and Multiplication Laws of Probability

Total Probability and Bayes Theorem

36. Descriptive Statistics

Describing Data

Exploring Data

37. Discrete Probability Distributions

Discrete Probability Distributions

The Binomial Distribution

The Poisson Distribution

The Hypergeometric Distribution

38. Continuous Probability Distributions

Continuous Probability Distributions

The Uniform Distribution

The Exponential Distribution

39. The Normal Distribution

The Normal Distribution

The Normal Approximation to the Binomial Distribution

Sums and Differences of Random Variables

40. Sampling Distributions and Estimation

Sampling Distributions

Interval Estimation for the Variance

41. Hypothesis Testing

Statistical Testing

Tests Concerning a Single Sample

Tests Concerning Two Samples

42. Goodness of Fit and Contingency Tables

Goodness of Fit

Contingency Tables

43. Regression and Correlation

Regression

Correlation

44. Analysis of Variance

One-way Analysis of Variance

Two-way Analysis of Variance

Experimental Design

45. Non-parametric Statistics

Non-parametric Tests for a Single Sample

Non-parametric Tests for Two Samples

46. Reliability and Quality Control

Reliability

Quality Control

47. Mathematics and Physics Miscellany

Dimensional Analysis in Engineering

Mathematical Explorations

Physics Case Studies

48. Engineering Case Studies

Engineering Case Studies

49. Student's Guide

Student's Guide

50. Tutor's Guide

Tutor's Guide