Calculus of Several Variables

Peter J. Eccles Lecture notes
Problem sheets
Diary of amendments to this webpage

Accessing files via this website

Clicking links on the page will lead to a request for a username and password. Access requires your usual University username and password. In case of difficulty let me know.

Lecture times and places

Tuesdays 12 in Schuster Building Blackett Theatre

Thursdays 11 in Schuster Building Blackett Theatre

Tutorial feedback classes times and places

Students are expected to register for one of the tutorial classes and to attend each week. Please do not attend more than one class each week. If you need additional help then contact me. It is essential to work at the problems sheets provided and assistance in doing this will be provided in the tutorial classes.

Tuesdays 3 in Alan Turing Building Louis Mordell Lecture Room (G.209)

Wednesdays 11 in Zochonis Building Lecture Theatre D

Wednesdays 12 in Alan Turing Building Louis Mordell Lecture Room )G.209(

Additional problems are discussed in the feedback classes to help students with the Problem sheets. I will post here a brief record of the problems discussed.

Queries

Any queries or comments about this course can be emailed to me at pjeccles@manchester.ac.uk. I will try to reply within 48 hours.

Office Hours

Students are welcome to call on me in my office (Alan Turing 1.111) at any time. If I am busy I will ask you to come back at another time and if possible arrange one. If I am not in my office then leave a not under my door or send me an email saying when you will return. I will try to make a point of being in my office available to students at the following time.

Tuesdays, 1 to 3 p.m.

However, sometimes I will have other commitments at this time.

Student Feedback

Books

See reading lists for more information.

It appears difficult to find a book which covers the material in this course at a similar level. The most comprehensive book is the following.

Wendell Fleming, Functions of Several Variables

This covers most of the material in the course but is pitched at a slightly more advanced level than the course. It is the book I have found most useful in preparing the course.

The following free book on the internet may be useful (the file is quite large and so takes a few second to load).

William Trench, Introduction to Real Analysis

Chapters 5 and 6 cover most of the material in the course except for the later material on differential forms and integration, again pitched at a slightly more advanced level than the course.

A slightly less advanced book which covers some of the material in the course is the following.

Seán Dineen, Multivariate Calculus and Geometry

The first four chapters of this book are useful for the first few weeks of the course.

Course notes

Background material

This lecture builds on material from several previous courses in particular MATH20101/20111 Real Analysis, MATH10202/10212 Linear Algebra and MATH10122/10232 Calculus and Vectors. For convenience most background material which is used is summarized in background notes posted on this website. Some of this material will be referred to in lectures, usually displayed by overhead projector. This material is not examinable except when it forms a part of results discussed in the lectures. (These links are not yet working.)

Lecture notes

I will aim to post these prior to the lectures but may sometimes post revised versions after the lectures in the light of student feedback. The titles are provisional for notes not yet posted. I will refer to these on-line notes in the lectures and it is useful to have them available to annotate in the lectures.

1. Continuous Functions of Several Variables
2. Differentiation of Real-Valued Functions of Several Variables
3. Differentiation of Vector-Valued Functions of Several Variables
4. Critical Points and Higher Derivatives
5. Differential Forms and the Integration of Differential Forms

Problem sheets

Solution sheets

Solutions sheets will be posted after students have had time to work at the problems.

Mid-semester coursework

The coursework will be an in-class test on Thursday 19 April (at the usual lecture time) contributing 15% of the marks for the assessment of this course unit.

Examination

The remaining 85% of the marks come from the end of semester examination. The rubric in 2012 will be the same as in 2010 and 2011 as follows.

Answer all four questions in Section A (40 marks in all) and three of the four questions in Section B (15 marks each). If all four questions from Section B are attempted then credit will be given for the three best answers only. The total number of marks on the paper is 85. A further 15 marks are available from work during the semester making a total of 100.

Here is some feedback on the June 2009 examination.

Here is some feedback on the June 2010 examination.

Previous papers

It should be noted that the lecturer for this course unit was changed in 2008-2009 and, although the syllabus is essentially the same, the style of the course unit is rather different. Here are some comments on the 2008 examination.

Diary of amendments to and additions to this webpage

6 January 2012: Initial posting of website (no links working yet).

28 January 2012: Notes A, B and C posted.

28 January 2012: Notes 1 posted.

28 January 2012: Problems 1 posted.

30 January 2012: Change to date of coursework test posted (now 19 April).

4 February 2012: Week 1 tutorial exercises posted.

4 February 2012: Time and place of additional tutorial posted.

5 February 2012: Notes 2 posted.

5 February 2012: Problems 2 posted.

11 February 2012: Week 2 tutorial exercises posted.

11 February 2012: Problems 3 posted.

11 February 2012: Amended Notes 1 posted (minor correction to the statement of Proposition 1.6).