DE1-MEM: Engineering Mathematics

Dr Sam Cooper - Dyson School of Design Engineering

Contents

Front Matter

• About the Course
• Course Support and Assessment
• Further Resources
• Licence

Chapter 0 - Refresher

• 0.1 - Algebra
• 0.2 - Calculus
• 0.3 - Using calculus
• 0.4 - Powers, logs & bases
• 0.5 - Engineers Love

Chapter 1 - Functions

• 1.1 - Curve Sketching
• 1.2 - Symmetry of functions

Chapter 2 - Vectors

• 2.1 - Co-ordinate Geometry
• 2.2 - Vector multiplication
• 2.3 - Vector equation of line

Chapter 3 - Matrices

• 3.1 - Matrix Operations
• 3.2 - Rules of Addition and Multiplication
• 3.3 - Transpose
• 3.4 - Square Matrices
• 3.5 - Inverses
• 3.6 - Linear Systems
• 3.7 - Labels

Chapter 4 - Linear Transformations

• 4.1 - Demystifying Linear Transformations
• 4.2 - One Dimension
• 4.3 - Two Dimensions
• 4.4 - Three Dimensions
• 4.5 - Determinant and Inverse

Chapter 5 - Eigenproblems

• 5.1 - Definitions
• 5.2 - Calculating Eigensolutions
• 5.3 - Finding All Eigenvalues
• 5.4 - Finding All Eigenvectors
• 5.5 - Interpretation Of Eigensolutions

Chapter 6 - Sequences and Series

• 6.1 - Sequences
• 6.2 - Series
• 6.3 - Limits and Convergence
• 6.4 - Truncated sum of 1, $n$, $n^2$ and $n^3$
• 6.5 - Mind Blown

Chapter 7 - Power Series

• 7.1 - Maclaurin Series
• 7.2 - Taylor Series

Chapter 8 - Complex Numbers

• 8.1 - Operations with complex numbers
• 8.2 - Finding complex roots
• 8.3 - De Moivre’s Theorum
• 8.4 - Imaginary numbers really exist

Chapter 9 - Ordinary Differential Equations

• 9.1 Back to basics
• 9.2 A function which is it’s own derivative
• 9.3 Categories
• 9.4 ODEs in physical systems
• 9.5 ODE’s summary

Chapter 10 - Coupled Oscillators

• 10.1 Sum of forces
• 10.2 Natural frequencies and Eigenmodes
• 10.3 Example system
• 10.4 Generalising
• 10.5 Mind blown

Chapter 11 - The Laplace Transform

• 11.1 Origins of the Laplace transform
• 11.2 But what does it mean?
• 11.3 Finding Laplace Transforms
• 11.4 Solving ODEs and ODE systems

Chapter 12 - Fourier Series

• 12.1 Periodic functions
• 12.2 Complex exponential representation
• 12.3 Fourier transform
• 12.4 Mind blown

Chapter 13 - Multivariate Calculus

• 13.1 Functions of multiple variables
• 13.2 Partial derivatives
• 13.3 Stationary points
• 13.4 Total differentials and derivatives
• 13.5 Vector calculus

Chapter 14 - Partial Differential Equations

• 14.1 Recap
• 14.2 Introducing PDEs
• 14.3 Separation of Variables
• 14.4 Diffusion Equation
• 14.5 Further Examples
• 14.6 Initial Conditions
• 14.7 Boundary Conditions
• 14.8 Outlook

Chapter 15 - Finite Differences

• 15.1 Introduction
• 15.2 Application Example - Numerical Diffusion
• 15.3 Systems of equations and conditions
• 15.4 Notation
• 15.5 Code

Chapter 16 - Root Finding

• 16.1 The Bisection Method
• 16.2 The Newton-Raphson Method
• 16.3 Secant Method

Chapter 17 - Optimisation

• 17.1 Linear Regression
• 17.2 Non-Linear Regression
• 17.3 Conclusion

Chapter 18 - The Normal Distribution

• 18.1 The Gaussian Integral
• 18.2 The Normal Distribution

If you are a converting the remaining topics from the pdf to this webpage, here are some things to look out for.

About the Course

This course is a rapid introduction (or reminder for some) to a range of topics that you will find useful during your engineering career. A huge amount of wonderful resources have become freely available online in the past few years, in the form of videos, blogs, forums, wikis etc.. My hope for this course is that you finish with the confidence necessary to look up questions that you don’t understand and hopefully re-purpose methods from one area to another.

Some undergraduate courses expect students to memorise a lot of formulae and derivations; however, now that all of mankind’s collected knowledge is just a few clicks away, there is no longer much value in this! Instead, we will focus on developing an intuitive understanding of the various topics, which I hope will not only be more useful, but also much more enjoyable and satisfying!

These notes are not intended to be comprehensive (that is what the internet is for), but instead hope to offer a fast paced and engaging description of the concepts, pitched at a level appropriate to DE1. Some the material is based on notes developed by Dr Rhazaoui, who taught the first iteration of this course.

Course Support and Assessment

Learning maths is a very personal activity, with each student having their own approach; however, to really understand what’s going on, there is no way around putting in the work on your own, occasionally getting stuck and thinking your way out. That said, I really hope the notes, lectures, online videos, tutorial sheets and quizzes help to push you in the right direction and keep you motivated!

Every week, you will take a short non-credit quiz to help me (and you) understand how you’re getting on. The course will be assessed through a combination 4 progress tests at half termly intervals, as well as 2 more substantial exams at the beginning of terms two and three. The course is two terms long and each week we will have 2 one hour lectures introducing the material. We will also have weekly tutorial sessions which will be 2 hours in the first term and 1 hour in the second. These sessions are primarily intended for you to ask the tutors questions about the material from the previous weeks and are not ideal for quite study. We will use Learning Catalytics to support the learning process, by running live quizzes.

Further Resources

KL Stroud and DJ Booth, Engineering Mathematics, 7th Ed., Macmillan, 2013 (Imperial library 510.246STR), is probably the core text for 1st year Maths, although ML Boas, Mathematical Methods in the Physical Sciences, 3rd Ed., Wiley, 2006 (Imperial library: 530.15BOA) is a bit less wordy and goes into some more advanced topics as wells.

WolframAlpha is a brilliant mathematical resource and if you are ever stuck with a question, this should be one of your first ports of call. Finally, I would like to recommend several wonderful YouTube series, including WelchLabs, 3Blue1Brown, blackpenredpen and Numberphile as a source of mathematical inspiration and delight.

Licence

This work is licensed under the Creative Commons Attribution- Noncommercial- Share Alike 2.0 UK: England \& Wales License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

These notes were written by Dr Sam Cooper and Dr Freddie Page of the Dyson School of Design Engineering, Imperial College London - corrections to samuel.cooper@imperial.ac.uk.

Thanks to Dr Khalil Rhazaoui for bravely pulling together the first draft of the notes for this course, which were helpful as a basis for creating later versions. Thanks also to Prof. Dave Dye who mentored me as a lecturer and whose notes were helpful for some of the later chapters of this book.