20411 Online Resources (2015-2016)
Lecturer: Dr. Catherine Powell
Email: c_dot_powell_at_manchester_dot_ac_dot_uk
Office: 1.124, Alan Turing Building.
Office Hours: Monday 3-4pm, Tuesday 3-4pm.
At the above times, I am guaranteed to be in my office. Students may see me at those times to discuss MATH20411 without an appointment. If you want to meet me at another time, just send me an email to arrange it.
Syllabus
The course syllabus is available from the School of Mathematics web site.
Lectures and Examples Classes
- Lectures: Monday 12-1, Tuesday 12-1, Schuster Building, Bragg Lecture Theatre.
- Examples Classes: Monday 1-2, Tuesday (1-2), Alan Turing Building G.205. Students should only attend the class allocated to them.
Online Lecture Notes
I will post summaries of lecture notes (roughly every two weeks) below. These should be used to supplement your own notes taken in lectures. To actively engage with the material in the course, students must also take their own notes. Passively reading online notes is no substitute for actually doing mathematics.
Students who have special learning plans, as directed by the DSO, should contact me at the start of term to make any necessary arrangements.
- Section 1: Introductory Material
- Section 2: Fourier Series
- Section 3: Introduction to PDEs
- Section 4: Separation of Variables
- Section 5: Finite Difference Methods
- Section 6: Vector Calculus
Handouts
Additional material in the form of handouts, to read between lectures, will be made available online below. I will usually also give out paper copies of these at the end of lectures.
- Week 0: Classical PDEs (This gives an overview of some of the PDEs you will meet in the course).
- Week 2: Orthogonal Vectors
- Week 3: Fourier Series
- Week 5-6: Separation of Variables
- Week 7: Bessel functions
- Week 8: Centered Finite Differences for Reaction-Diffusion Equation
- Week 8: Finite Differences for Convection-Diffusion Equation
- Week 9: Explicit Finite Differences for 1D Heat Equation
- Week 10: Implicit Finite Differences for 1D Heat Equation
- Week 10:Vectors and Div,Grad,Curl
- Week 11-12: Line integrals (revision)
Example Sheets
On average, there is one example sheet per week. However, questions are grouped according to topics. Therefore, the numbering of the examples sheets will not necessarily correspond to the week number. This is no reason to panic. Below, I will indicate which questions you should attempt in which weeks.
- Sheet 1: Introductory material and solutions.
- Sheet 2: Orthogonality (Attempt questions 1-5 before week 3 examples class).
- Sheet 3: Fourier Series
- Sheet 4: Partial Differential Equations
- Sheet 5: Separation of Variables A
- Sheet 6: Separation of Variables B
- Sheet 7: Finite Difference Methods A
- Sheet 8: Finite Difference Methods B
- Sheet 9: Vectors and Multiple Integrals
- Sheet 10: Vector Calculus
Some sheets will certainly have more questions than you will be able to do in one week, but these can be used for revision later. Example classes start in week 2. There will be many students in the examples class, so you will get the most benefit out if the class of you try the questions beforehand. You can then ask questions about the problems you are unable to do. You should attend one example class per week.
Numerical Analysis
The material in Section 5 on finite difference methods will give students their first taste of numerical analysis. This is a branch of applied mathematics with many important practical applications in the real world. For more details, and a list of other numerical analysis courses, see the Numerical Analysis undergraduate student pathway.
MATLAB
Students will be required to use MATLAB occasionally and should know how to set up vectors, perform mathematical operations on vectors, write simple programmes and plot functions. Demos will be given in examples classes throughout the term and examples given on handouts. Useful MATLAB resources and tutorials can be found on the web, including, HERE. An extensive range of MATLAB manuals are also available at the library.
A useful summary of basic commands is given here: MATLAB essentials
For certain lectures (e.g. the ones on finite difference methods in weeks 9 and 10) students will need the following MATLAB codes. Download the files and save to your P-drive. Open them in the MATLAB editor and read the instructions.
- fourierN_demo.m (illustrates convergence of a Fourier Series - example from Fourier Series notes)
- heateq_demo.m (plots the solution to the Heat Equation - example from Classical PDEs notes)
- trisolve.m (solves a tridiagonal system of equations)
- reac_diff_1d.m (implements centred finite differences for the reaction-diffusion problem)
- conv_diff_1d.m (implements centred finite differences for the convection-diffusion problem)
- heat_eq_explicit_fd.m (implements the explicit finite difference scheme for the heat equation)
Textbooks
You do not need to buy any textbooks for this course . We will study several basic topics in calculus and applied mathematics, which are covered in hundreds of available texts in the library. However, the following books all contain some material you will meet in the course.
- James Stewart, Calculus, Early Transcendentals, Thomson, fifth edition (international student edition), 2003.
- (Useful for the first part of the course and vector calculus).
- R Haberman, Elementary Applied Partial Differential Equations with Fourier
Series and Boundary Value Problems, (Third edition) Prentice-Hall, 1998.
- (Useful for the section on Fourier Series and introduction to PDEs).
- Morton, K.W., Mayers, D.F, Numerical solution of partial differential equations, Cambridge University Press, 2005.
- (Useful for the sections of finite difference methods and numerical analysis).
- Schey, H. M. Div, Grad, Curl, and all that : an Informal Text on Vector Calculus, New York : W. W. Norton, various editions.
- (Useful for the final few weeks of the course when we tackle vector calculus).
Coursework
The coursework test will take place in Reading week, on Monday 2nd November at 1pm, in the Simon Building, Theatre B.
Exam Revision Resources
Past papers are available from the main School of Mathematics website . Apart from the sample paper, solutions to exam papers are not provided. Solutions to examples sheets and the sample exam will help you revise.
The following documents provide feedback on general performance on past exam papers and common mistakes.