HELM : Helping Engineers Learn Mathematics. A resource of .pdf files covering many different topics.
Section 1 : Basic Algebra
- 1.1 Mathematical Notation and Symbols
- 1.2 Indices
- 1.3 Simplification and Factorisation
- 1.4 Arithmetic of Algebraic Fractions
- 1.5 Formulae and Transposition
Section 2 : Functions
- 2.1 Basic Concepts of Functions
- 2.2 The graph of a function and parametric form Note that, bottom of page 13, the domain stated as involving ‘x’ should be written in terms of ‘t’ instead.
- 2.3 One to one and inverse functions
- 2.4 Characterising Functions
- 2.5 The Straight Line
- 2.6 The Circle
- 2.7 Some Common Engineering Functions
Section 3 : Polynomials, Inequalities and Partial Fractions
- 3.1 Solving Linear Equations
- 3.2 Solving Quadratic Equations
- 3.3 Solving Polynomial Equations
- 3.4 Solving Simultaneous Linear Equations
- 3.5 Solving Inequalities
- 3.6 Partial Fractions
Section 4 : Trigonometry
- 4.1 Right-Angled Triangles
- 4.2 Trigonometric Functions
- 4.3 Trigonometric Identities
- 4.4 Applying Trigonometry to Triangles
- 4.5 Applying Trigonometry to Waves
Section 5 : Functions and Modelling
- 5.1 The Modelling Cycle and Functions
- 5.2 Quadratic Functions and Modelling
- 5.3 Oscillating Functions and Modelling
- 5.4 Inverse Square Functions and Modelling
Section 6 : Logarithms and exponentials
- 6.1 The exponential Function
- 6.2 The hyperbolic function
- 6.3 Logarithms
- 6.4 The logarithm function
- 6.5 Log-linear graphs
- 6.6 Modelling Exercises
Section 7 : Matrices
Section 8 : Using matrices and determinants to solve equations
- 8.1 Cramer’s rule for solving simultaneous equations
- 8.2 Solving simultaneous equations using the inverse matrix
- 8.3 Gauss elimination
Section 9 : Vectors
- 9.1 Basic concepts of vectors
- 9.2 Cartesian components of vectors
- 9.3 The Scalar Product
- 9.4 The Vector product
- 9.5 Vectors, Lines and Planes
- 9.6 Vectors and Electrostatics
Section 10 : Complex numbers
- 10.1 Complex arithmetic
- 10.2 Argand diagrams and polar form
- 10.3 Exponential form
- 10.4 De Moivre’s theorem
Section 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
Section 12 : Applications of differentiation
- 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
Section 13 : Integration
- 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 by Partial Fractions
- 13.6 Integration of Trigonometric Functions
Section 14 : Applications of Integration I
- 14.1 Integration of the Limit of a Sum
- 14.2 Mean Value and RMS Value
- 14.3 Volumes of Solids of Revolution
- 14.4 Lengths of Curves and Areas of Surfaces of Revolution
- 14.5 Integration by Substitution and using Partial Fractions
Section 15 : Applications of integration II
Section 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
Section 17 : Conic sections
- 17.1 Conic sections (circle, ellipse, parabola and hyperbola) Note that the answer to question 6 on page 22 is incorrect
- 17.2 Polar co-ordinates
- 17.3 Parametric curves
Section 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
Section 19 : Differential equations
- 19.1 Modelling with differential equations
- 19.2 First Order Ordinary Differential Equations
- 19.3 Second Order Ordinary Differential Equations
- 19.4 Applications of Differential Equations
Section 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
Section 21 : z-Transforms
- 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 Basic Concepts
- 22.2 Applications of Eigenvalues and Eigenvectors
- 22.3 Repeated Eigenvalues and Symmetric Matrices
- 22.4 Numerical determination of Eigenvalues and Eigenvectors
Section 23 : Fourier Series
- 23.1 Periodic Functions
- 23.2 Representation of 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 Applications of Fourier Series
Section 24 : Fourier Transforms
- 24.1 The Fourier Transform
- 24.2 Properties of the Fourier Transform
- 24.3 Some Special Fourier Transform Pairs
Section 25 : Partial Differential Equations
- 25.1 Partial Differential Equations
- 25.2 Applications of PDEs
- 25.3 Separation of Variables
- 25.4 Solution by Fourier Series
Section 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
Section 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
Section 28 : Differential Vector Calculus
- 28.1 Background to Vector Calculus
- 28.2 Differential Vector Calculus
- 28.3 Orthogonal Curvilinear Coordinates
Section 29 : Integral Vector Calculus
- 29.1 Line Integrals Involving Vectors
- 29.2 Surface and Volume Integrals
- 29.3 Integral Vector Theorems
Section 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
Section 31 : Numerical Methods of Approximation
- 31.1 Polynomial Approximation
- 31.2 Numerical Integration
- 31.3 Numerical Differentiation
- 31.4 Non-linear 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
Section 33 : Numerical Boundary Value Problems
Section 34 : Modelling Motion
Section 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
Section 36 : Descriptive Statistics
Section 37 : Discrete Probability Distributions
- 37.1 Discrete Probability Distributions
- 37.2 The Binomial Distribution
- 37.3 The Poisson Distribution
- 37.4 The Hypergeometric Distribution
Section 38 : Continuous Probability Distributions
- 38.1 Continuous probability distributions
- 38.2 The uniform distribution
- 38.3 The Exponential Distribution
Section 39 : The normal distribution
- 39.1 The random distribution
- 39.2 The normal approximation to the binomial distribution
- 39.3 Sums and differences of random variables
Section 40 : Sampling Distributions and Estimation
Section 41 : Hypothesis Testing
Section 42 : Goodness of Fit and Contingency Tables
Section 43 : Regression and Correlation
Section 44 : Analysis of Variance
Section 45 : Non-parametric Statistics
Section 46 : Reliability and Quality Control
Section 47 : Mathematics and Physics Miscellany
Section 48 : Engineering Case Studies
Section 49 : Students’ Guide
Section 50 : Tutor’s Guide