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School of Mathematics

# 2M1 Course Materials

Monday Lectures weeks 1-5 ex-M stream

2M1: Functions of two variables and PDEs

# 2M1: Functions of two variables and PDEs

This part of the course, dealing with functions of two variables and partial differential equations (PDEs), is taught during weeks 5,7,8,9 and 10 in Renold H11 on Monday 11 a.m. by Prof. Matthias Heil. This page provides online access to the lecture notes, example sheets and other handouts and announcements. Most of the material will be taught in "chalk and talk" mode. If OHP transparencies are used, copies will be made available (after the lecture) on this page. Please consult the service course page for details on how to get hold of material for the other parts of the course.

Please note that the lecture notes only summarise the main results and will generally be handed out after the material has been covered in the lecture. You are expected take notes during the classes.

NOTE: The various links on this page (probably) won't work from the mirrored version on the school's service course pages. Please go to the original at http://www.maths.man.ac.uk/~mheil/Lectures/2M1/index.html before trying to download any of the handouts.

## Syllabus:

• Functions of two variables and practical examples. Quick reminder of ordinary and partial derivatives; reminder of Taylor expansions and maxima/minima for functions of one variable. Generalisation to 2D. Criteria for minima/maxima/saddle points. 2D Taylor series.
• What are PDEs? Where do they arise? What are their solutions? Examples and their physical/engineering origin: 1D advection, 2D Laplace, 1D unsteady heat and 1D linear wave equations. Boundary and initial conditions. How to verify that a function is a solution of a PDE. Solving PDEs by separation of variables.

## Do-able questions

 Date Topics covered Do-able questions 26/10/09 Review of functions of a single variable (plots; derivatives; stationary points; Taylor series). Motation for fcts of multiple (two) variables; partial derivatives; plots; necessary conditions for a stationary point. Example Sheet I: Q1 and the first half of Q2. 09/11/09 Classification of stationary points; 2D Taylor series. Example. All questions on Example Sheet I. 16/11/09 Reminder of ODEs: What are they? BC/IC and BVP/IVP. Physical examples. What are PDEs? How do we check if a candidate solution solves a given BVP/IVP? Examples of PDEs, their physical background and suitable BC/IC: 1D Advection, 2D Laplace, 1D unsteady heat, 1D linear wave. Question 1 on Example Sheet II. 23/11/09 Separation of variables: A step-by-step guide, illustrated for the linear wave equation. All questions on Example Sheet II. 30/11/09 Separation of variables for the unsteady heat equation. All questions on Example Sheet II.

## Assessment:

The course will be examined in a two hour exam in January/February 2010.

Please note a few corrections for previous handouts (the files above have already been corrected).

• Maxima and Minima. Some interactive diagrams of maxima and minima.
• Monday Lectures weeks 1-5 ex-P stream

## Partial Differential Equations Materials

These lectures are taken only by students who took the course units 1P1 and 1P2 in year 1.

Partial Differential Equations occur in many aspects of Mechanical and Aerospace Engineering. These five lectures will consider

• 1. The concept of a partial differential equation
• 2. Several specific partial differential equations i.e. the wave equation, the heat equation and Laplace's equation
• 3. Solution of PDEs by separation of variables
• 4. Solution of first-order PDEs by the method of characteristics
• 5. Solution of certain second-order PDEs by the method of characteristics
• Course notes. These are condensed versions of the notes presented in lectures and are not a substitute for constructive attendance at lectures.

### Handouts

Copies of handouts given in lectures

### Progress through example sheets

By the tutorials on the dates indicated, students should have made serious attempts at the questions marked

• Tuesday 3rd October (week 2). Sheet 1, Questions 1-5
• Tuesday 17th October (week 4). Sheet 2, All Questions
• Tuesday 7th November (week 7). Sheets 3 and 4, All Questions

Wednesday Lectures weeks 1-9

## Statistics Materials

### Couresework

Monday Lectures weeks 7-10, Wednesday lecture week 10

## Multiple and Line Integrals

In many areas of Mechanical or Aerospace Engineering, it is necessary to evaluate a multiple integral or a line integral. A double integral represents the volume under a surface but above a particular rectangle or other shape. A line integral can represent an area above a curve but beneath a function (or the projection of this onto a particular axis. These five lectures will consider

• 1. Double integrals over rectangular areas.
• 2. Double integrals over non-rectangular areas and assembling such an area from smaller areas.
• 3. Double integrals achieved by change of variables.
• 4. The Geometry of Line integrals.
• 5. Evaluation of Line integrals.
• 6. Matters concerned with the contour of integration.

### Handouts

Copies of handouts given in lectures

### Progress through example sheets

By the tutorials on the dates indicated, students should have made serious attempts at the questions marked

• Tuesday 21st November. Sheet 5 : Questions 1-4
• Tuesday 5th December. Sheet 6 : All Questions

weeks 11-12

## Laplace Transforms

### Handouts

Copies of handouts given in lectures

## HELM Modules

Download and print the HELM modules on topics relevant to 2M1.

Basic Probability (context material)
Sets
Elementary probability
Addition and multiplication laws of probability (including permutations & combinations)
Total Probability and Bayes' Theorem
Statistics
Discrete Probability Distributions
The Binomial Distribution
The Poisson Distribution
The Hypergeometric Distribution
Continuous probability distributions
The uniform distribution
The Exponential Distribution
Functions of two variables
Functions of several variables
Partial derivatives
Stationary points
Errors and percentage change
Partial Differential Equations
Partial Differential Equations
Applications of PDEs
Separation of Variables
Solution by Fourier Series
Multiple integration
Introduction to Surface Integrals
Multiple Integration over Non-rectangular regions
Triple Integrals
Changing Coordinates
Laplace Transforms
Causal functions
The transform and its inverse
Further Laplace transforms
Solving differential equations
The convolution theorem
Transfer functions

### Formula Tables

Download, print and get used to looking up information in these Formula Tables. Your lecturer will be able to confirm if these tables will be provided in exams.

## Exam papers

The January 2007 Exam will last two hours and will consist of five questions. Each student will answer THREE of the five questions. The questions are as follows

• Question 1 : On Dr Heil's work during the first 5 Mondays
• Question 2 : On Dr Steele's work during the first 5 Mondays
• Question 3 : Statistics
• Question 4 : Double integrals and line integrals
• Question 5 : Laplace Transforms

Download and print exam papers from previous years. Be aware that the syllabus may have changed, making some questions no longer relevant to the course you are taking. The number of questions may also have changed.

Unofficial Solutions to some questions from

Last modified: September 28, 2010 9:13:04 AM BST.