# Maths Refresher

We are currently updating this course, and the new version will be available by the end of summer 2023

## Introduction

This material assumes familiarity with algebra at the non-calculus level, and with elements of two-dimensional geometry. Different topics are summarised in separate sections and worked examples are given to show the application of the material. These are followed by problems which the student should undertake to ensure satisfactory understanding of the section.

Many students may already know much of the material presented, in which case it may only be necessary to make a quick reminder of the terminology used throughout the course. It may also be useful for access to formulae, and several oft-used relationships, differentials and integrals are summarised in an appendix. More advanced mathematical material than appears here may be introduced as needed during the course. Definitions of mathematical terms are printed in bold type.

## 1. Functions

On revision of this section and completion of these exercises you should be able to:

• define the concept of the function
• calculate the values of functions of a single variable
• define scalar and vector variables
• multiply vectors to give scalar and vector products

## 2. Differentiation

On revision of this section and completion of these exercises you should be able to:

• define differentiation
• calculate the differentials of simple functions
• describe the derivation of the base of natural logarithms and the natural log of x
• determine the first differential and first derivative of a range of complicated functions

## 3. Integration

On revision of this section and completion of these exercises you should be able to:

• define integration
• integrate functions of a single variable
• evaluate integrals which are a functions of two variables (double integration)
• describe the principles of triple integration

## 4. Complex numbers

On revision of this section and completion of these exercises you should be able to:

• define the following terms in relation to one another: complex number, modulus, argument
• add, subtract, multiply and divide using complex numbers

## 5. Trigonometric functions

On revision of this section and completion of these exercises you should be able to:

• define the 6 key ratios of trigonometry: sin, cos, tan, cosec, sec, cotan
• manipulate trigonometric functions to solve equations
• differentiate and integrate trigonometric functions
• use the exponential function to validate equations
• write complex numbers in polar form

## 6. Series

On revision of this section and completion of these exercises you should be able to:

• define the maths term ‘series’
• illustrate how finite series are derived
• express exponential functions as series
• define the binomial theorem
• use binomial expansion to solve problems
• define Taylor expansion
• use Taylor expansion to solve problems.

## 7. Differential equations

On revision of this section and completion of these exercises you should be able to:

• describe the relationship expressed by differential equations
• define the following terms in relation to differential equations: linear, non-linear, first order, second order
• apply the above principles to solve equations
• use a knowledge of physics to test solutions to equations.

## 8. Partial differentiation

On revision of this section and completion of these exercises you should be able to:

• define the partial differentiation of a function comprised of two variables
• define the partial differentiation of a function comprised of three variables
• use partial differential equations to solve closed problems

## 9. Coordinate systems

On revision of this section and completion of these exercises you should be able to:

• define Cartesian coordinates
• define vectors in terms of Cartesian coordinates
• perform calculations using Cartesian coordinates
• define spherical polar coordinates
• perform calculations using spherical polar coordinates
• define cylindrical polar coordinates
• perform calculations using cylindrical polar coordinates

## 10. Probabilities and errors

On revision of this section and completion of these exercises you should be able to:

• define the probability distribution function
• define and calculate the values: mean, variance and standard deviation
• define the Gaussian distribution
• define the Poisson distribution
• calculate the probability of events within a Poisson distribution
• describe the circumstances in which a Poisson distribution may approach a Gaussian curve
• define the term error and explain how errors are propagated
• calculate the standard error of the mean
• explain how systematic errors can be accommodated when quoting measurements
• use least squares fitting to determine the slope of the line of best fit.

## 11. Mathematical Formulas

This section gives the mathematical formulas for:

• Exponentials and logarithms
• Solid bodies
• Vectors
• Differentiation and integration
• Product rule for differentiation
• Quotient rule for differentiation
• Integration by parts
• Complex numbers
• Trigonometric functions
• Series
• Geometric series
• Binomial expansion
• Taylor Series
• Differential equations
• Partial differentiation
• Coordinate systems
• Cartesian coordinates
• Spherical polar coordinates
• Cylindrical polar coordinates

The following resources may be useful as further references. They’re presented in order of usefulness.

Mathcentre student website
https://www.mathcentre.ac.uk/students.php Leaflets available on topics e.g. functions, complex numbers, trigonometric functions etc. with non-interactive exercises (answers provided). Some but not all of the materials appear to be the same as the HELM resources.

Booklets containing more detail with exercises and answers on similar topics such as Cartesian components of vectors and trigonometric functions.

Exercise booklets on a limited range of topics containing lots of exercises with answers e.g. 29 pages of problems on differentiation should you require them!

Diagnostic tests on a big range of topics. These use Active X controls which you may need to download. These resources don’t seem to have been used by many students according to their ‘ratings’ so their quality is unconfirmed.

Short video clips : an excellent range of clips illustrated by examples worked on stream. You’ll need Windows Media Player 9 or equivalent and sound to get the benefit.

Online DVDs with detailed coverage of algebra, differentiation, integration and trigonometry.
Search function: if you can’t find what you want in the above, you can search the site.

University of Plymouth Maths Support Website
https://www.plymouth.ac.uk/schools/school-of-engineering-computing-and-mathematics/mathematical-sciences/mathematical-sciences-outreach/mathaid

Some good materials here in .pdf format with interactive exercises and solutions. Topics include vectors and differentiation.

BBC maths education website
https://www.bbc.co.uk/teach/skillswise/maths/zfdymfr

Some excellent first principle explanations of e.g. differentiation and integration here. Not an interactive site but maybe a good starting point if you’re really rusty!

Helping Engineers Learn Mathematics (HELM) website
https://www.lboro.ac.uk/departments/mlsc/student-resources/helm-workbooks/

Two types of resources might be useful: Workbooks and CAL packages.

Workbooks: full contents are not available. However, if you click on a topic, sections which are underlined can be accessed as .pdfs. e.g. some aspects of functions, equations and complex numbers.

CAL courseware: this covers 3 areas: sequences and series, matrix multiplication partial differentiation. To use the courseware you need to download Macromedia Authorware Player which you can get from the site. Some of the work may be too simple and the animations are a little irritating!