Geometry II: Knots and surfaces, MTH5109, 2011-12
Last updated 14th May 2012
Lectures commenced on Monday 9th January, at the following times and places:
Monday 9.00 - 10.00 Arts 1 Lecture Theatre
Wednesday 12.00 - 13.00 Arts 1 Lecture Theatre
Friday 14.00 - 15.00 FB 240
Examples classes were on Tuesdays 11.00 Arts 1 1.128 and Fridays 12.00 GO Jones U G1 (L G1 for overflow). These are now over.
I may also make here some administrative announcements:
Uncollected midterm scripts, old cwks and cwk 10 scripts (which will be marked in the week before Easter) can all be collected from the Maths office 101.
I have regular office hours now till the exam, hours will be posted on my web page.
I have uncharacteristically typed up my revision lecture(s) and these are downloadable here . Note that revision lectures can only outline a few key elements of the module and are not a replacement for the full set of lecture notes. Good luck!
This course is an introduction to the geometry of curves and surfaces
in three-dimensional space. We consider first the topology of simple
closed curves (knots). This is followed by discussion of properties
of curves which are metrical (i.e. depend on distances), such as
arc-length, normal, and curvature. Additional properties are
discussed for curves in a plane. Then we move on to surfaces, and
after discussing a few topological properties such as orientability,
we concentrate again on metrical properties, such as the fundamental
forms, curvature and geodesics. Some of these involve the behaviour
of curves lying in the surfaces. We conclude this part with the
Gauss-Bonnet theorem. As a non-examinable topic, if time, we shall discuss hyperbolic
spaces and/or higher-dimensional surfaces.
The techniques needed here are primarily differentiation and
integration, possibly in 2 or three dimensions, and some basic algebra
such as determinants.
This page will cover:
The approved syllabus and learning outcomes can be found on the module
specification pages here and here respectively.
The recommended text for most of the course is:
A Pressley, Elementary Differential Geometry, Springer UMS 2000.
You can get another point of view from a new book:
C Bar, Elementary Differential Geometry, Cambridge University Press, 2010.
The knot theory section is not covered by a suitable text but I recommend Chapter 1 of:
Download Professor MacCallum's 2009 Notes
(Please be aware that I may not cover exactly the same material or in the same order as in Professor MacCaullum's notes particularly in the later chapters. The later chapters were intended only to supplement the recommended text in any case.)
You can get a more sophisticated point of view of knot theory from:
W.B.R. Lickorish, An Introduction to Knot Theory, Springer, 1997.
You may also find the following pointers and handouts useful:
I will put lecture notes here, but normally after a deliberate delay of a couple of weeks. This is so as not to substitute for your turning up to my lectures:
Lectures 1-3
Lectures 4-6
Lectures 7-12 (updated pages 45-46 on 6/2/12)
Lectures 13-17 (updated 15/2/12)
Lectures 23-27 (lectures 18-22 were midterm/revision/post mortem)
Lectures 28-34 (lectures 35-36 will be cwk and revision)
Midterm and final revision lectures (typed, 4 pages) .
The formal pre-requisites are Calculus II (which itself requires
Calculus I) and Geometry I. Some of the things you need to know are:
From Calculus I
Basic differentiation and integration (Thomas, chapters 3 and 5).
Trigonometric functions (mainly cosx and sinx). (Thomas 1.6, 3.4, 8.4 and 8.5).
Log, exp and hyperbolic functions (cosh and sinh etc). (Thomas
7.2, 7.3 and 7.8).
Integration by parts. (Thomas, section 8.2)
From Calculus II
Basic idea of limits and continuity. (Thomas chapter 2).
Partial derivatives. The Chain Rule. (Thomas 14.3, 14.4).
Tangent planes and differentials. (Thomas 14.5, 14.6).
Double integrals. (Thomas 15.1, 15.3, 15.7)
From Geometry I
Vectors in 2-space and 3-space, expressed as xi + yj +zk or as row or
column vectors. Addition of vectors. Length of vectors.
Vector and cartesian equations of a straight line in R2 and
R3.
Scalar multiple and scalar product of vectors in R2 and
R3. Cartesian equation of a plane in R3.
Vector products in R3.
Linear transformations in R2 and R3 and their
matrices. Characteristic equation, eigenvalues and eigenvectors,
trace. Addition and multiplication of 2x2 and 3x3 matrices. Inversion of
matrices in R2 and in R3. Determinants.
Cartesian equations of ellipse, parabola, hyperbola.
Problems sheets will be posted here as they are issued, and answers after the relevant hand-in date and I will also try to bring hard copies to the relevant lecture. The starred question will be marked with feedback but will not contribute to your final mark.
Solutions to the coursework should be handed in to the BLUE BOX IN THE BASEMENT by 4.15pm on the Friday indicated (this will generally be the week after the problem sheet was handed out) and returned in lectures. It will thereafter be collectable from the Maths. Office
Ex Sheet 1,
and answers to it.
Ex Sheet 2,
and answers to it.
Ex Sheet 3,
and answers to it (reloaded 15/2/12).
Ex Sheet 4,
and answers to it (reloaded 15/2/12).
Ex Sheet 5,
and answers to it.
Ex Sheet 6,
and answers to it.
Ex Sheet 7,
and answers to it.
Ex Sheet 8,
and answers to it.
Ex Sheet 9,
and answers to it.
Ex Sheet 10,
and answers to it.
The week 7 mid-term test counts 10% towards the final mark. Its format is broadly similar to last year with a 40 minute duration. Here is last years and its solution.
Last year's
test,
and answers to it.
This year's
test,
and answers to it.
The results profle was: 34% of you had 70-100% (including 9 of you 90% or more), 20% of you had 60-69%, 22% of you had 50-59%, 8% of you had 40-49% and 16% of you had <40%.
This is mostly good but with no room for complacency -- many of you got thru on the knot theory and this will have less weighting in the final exam, and the questions will also be a bit harder.
Many of you remain confused about dot and cross products and elementary calculus which are prerequsisites for the course and if this applies to you I ask you go over this at the tutorial classes where I and the support staff will help you.
If you have not already, please pick up your scripts from room 101 and go thru them with me or one of the helpers at tutorials. This is criticial for anyone who got <50% and highly recommended for everyone else.
Persistent non-submission of course- work or absence from lectures, exercise classes or tests may lead to you being barred from the final exam. If you miss handing in a coursework or taking the test for a good reason, you should fill in a Missed In-Term Assessment Report Form (available from the Web at http://www.maths.qmul.ac.uk/undergraduate/forms/ ) and discuss it with the Student Support Officer (Maths Office).
The date of the exam
will not be known until the college examination timetable is published. The general format of the exam will be the same as last year. Note in particular that since full marks in the exam requires complete answers to all questions, it will not be
possible to get a high mark without studying all parts of the
course as lectured. It also means that there may be more difficult questions or parts of
questions on different parts of the course in different years.
Note that the exam format changed in the 2009 exam so if you are revising
from previous years exams, please be aware of that.
The exam counts 90% towards the final mark. Previous years' papers can be found in the College's
online collection of Maths exam papers. Here are a couple.
2009
exam and
2009 exam
answers (please note that the Gauss-Bonnet was not covered that year, instead there was an extra qn on knot theory)
2011
exam (I am not releasing solutions as I want you to try the exam yourself after revising)
Please feel free to see me to discuss any problems that you are having
with the course, either in the form of a quesiton in lectures, at Examples classes or in my Office Hours. My department page here will show these and any special changes to the office hours as will, if possible, my office door.