MATH20132 Online Course Materials

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. If you get an error message then try refreshing the page. In case of difficulty let me know.

Lecture times and places

Tuesdays 12 in Schuster Building Bragg 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 James Lighthill Lecture Room (G.205)

Wednesdays 11 in Alan Turing Building Louise Mordell Lecture Room (G.209)

Wednesdays 12 in University Place Lecture Room 5.206

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.

• Week 1: Limits
  
• Week 2: Limits


• Week 3: Directional Derivatives


• Week 4: The Fréchet Derivative


• Week 5: Differentials and 1-Forms


• Week 6: Differentiating Vector-Valued Function


• Week 7: The Chain Rule and the Inverse Function Theorem


• Week 8: The Implicit Function Theorem


• Week 9: Lagrange Multipliers


• Week 10: Critical Points


• Week 11: Differential Forms


• Week 12: Exterior Differential and Surface Integrals

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.

Week 3 questionnaire feedback

I distributed week 3 feedback forms during the tutorials on 12 and 13 February. Here is some feedback from me on the forms.

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 which can be accessed electronically via the the above link to the University Library reading list.

• 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 seconds 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 course 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.

• A. Continuity of Real-Valued Functions of One Variable
  
• B. Differentiation of Real-Valued Functions of One Variable
  
• C. Linear Algebra

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. 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. Extreme Values, Critical Points and Higher Partial Derivatives


• 5. Differential Forms and the Integration of Differential Forms

Problem sheets

• 1. Limits and Continuity
  
• 2. The Directional Derivative


• 3. The Fréchet Derivative


• 4. Calculating the Derivative


• 5. Differentiating Vector-Valued Functions and the Chain Rule


• 6. The Inverse Function Theorem and the Implicit Function Theorem


• 7. Finding Extrema Subject to a Constraint


• 8. Critical Points and Higher Derivatives


• 9. Differential Forms and Integration

Solution sheets

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

• 1.


• 2.


• 3.


• 4.


• 5.


• 6.


• 7.


• 8.


• 9.

Mid-semester coursework

15% of the marks fork this course unit come from an in-class test which will took place during the usual lecture hour on Tuesday 16 April, immediately after the Easter Break.

Here are some solutions to the coursework test.

Examination

The remaining 85% of the marks come from the end of semester examination. The rubric in 2013 will be the same as in 2010, 2011 and 2012 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 are some solutions to the June 2010 examination.

Here are some solutions to the June 2011 examination.

Here is some feedback on the June 2012 examination.

Diary of amendments to and additions to this webpage

15 January 2013: Initial posting of the website. (Not all the links are working yet. I will announce here when files are posted).

15 January 2013: Notes A, B and C posted.

15 January 2013: Notes 1 posted.

28 January 2013: Problems 1 posted.

31 January 2013: Week 1 tutorial notes posted.

31 January 2013: Amended Notes A posted (minor typing error in Definition A.1).

31 January 2013: Amended Notes 1 posted (minor clarification to Definition 1.2).

6 February 2013: Notes 2 posted.

7 February 2013: Week 2 tutorial notes posted.

7 February 2013: Information about the time of the mid-semester test posted.

8 February 2013: Problems 2 posted.

11 February 2013: Week 3 feedback form posted.

13 February 2013: Week 3 tutorial notes posted.

14 February 2013: Solutions 1 posted.

14 February 2013: Problems 3 posted.

14 February 2013: Feedback on week 3 questionnaires posted.

21 February 2013: Week 4 tutorial notes posted.

21 February 2013: Solutions 2 posted.

21 February 2013: Problems 4 posted.

26 February 2013: Notes 3 (first part) posted. (I will try to replace this by the complete version in the next few days.)

28 February 2013: Week 5 tutorial notes posted.

28 February 2013: Solutions 3 posted.

5 March 2013: Amended Solutions 2 posted (apologies - I posted a draft version in error).

5 March 2013: Problems 5 posted.

8 March 2013: Week 6 tutorial notes posted.

8 March 2013: Amended Notes A posted (adding a section on properties of continuous functions).

8 March 2013: Amended Notes C posted (adding a sction on upper triangular matrices).

10 March 2013: Amended Problems 5 posted (minor correction to Question 3).

10 March 2013: Notes 3 posted with additional material from Example 3.23 onwards added (and a minor amendment to the proof of Corollary 3.16(b)).

12 March 2013: Amended Problems 5 posted (I changed the wrong thing on 10 March).

14 March 2013: Week 7 tutorial notes posted.

14 March 2013: Solutions 4 posted.

18 March 2013: Notes 4 (first part) posted. (I will try to replace this with the complete version in the next few days.)

18 March 2013: Problems 6 posted.

20 March 2013: Week 8 tutorial notes posted.

20 March 2013: Information about the content of the in-class test posted.

8 April 2013: Solutions 5 posted.

9 April 2013: Solutions 6 posted.

15 April 2013: 2012 coursework test and solutions posted.

15 April 2013: Location of the coursework test posted.

16 April 2013: Coursework solutions posted.

17 April 2013: Problems 7 posted.

17 April 2013: Week 9 tutorial notes posted.

18 April 2013: Notes 4 (complete version) posted.

18 April 2013: Notes 5 (and last) posted.

18 April 2013: Past examination papers and solutions posted.

25 April 2013: Week 10 tutorial notes posted.

25 April 2013: Problems 8 posted.

25 April 2013: Problems 9 (and last) posted.

26 April 2013: Solutions 7 posted.

26 April 2013: Amended solutions to June 2010 examination posted (corrections to A3(c) and B7(b)).

3 May 2013: Solutions 8 posted.

3 May 2013: Week 11 tutorial notes posted.

8 May 2013: Week 12 tutorial notes posted.

8 May 2013: Solutions 9 posted.

8 May 2031: Amended further solutions to June 2010 examination posted (correction to A4).

8 May 2013: Amended solutions to June 2011 examination posted (correction to A4).

8 May 2013: Amended Tutorial notes week 11 posted (minor error in Question 2).

9 May 2013: Amended feedback on 2012 examination (minor error in the solution to Question 4(b)).

11 May 2013: Amended solutions to June 2011 examination posted (correction to definition the function f).

13 May 2013: Amended solutions 9 posted (correction to the solution to Question 3(d)).

14 May 2013: Amended Notes 4 posted (correction to Theorem 4.8).