24
Oct 2002
Complaint about speed and writing. Todays lecture
seemed at first impressions terrribly complex, because of the speed of
presentation and the writing on the board on the far right being
illegible(my right), I appreciate the noise level makes it difficult
and maybe irksome but as I am not the strongest mathematician I enjoy
getting a good set of notes, anyway keep up the good work and see you
tomorrow morning!!
Reply Thanks for reminding me about my
writing, it tends to get worse and worse unless I am reminded. I am
sorry you found this lecture difficult, these calculations appear
formidable at first, but if you go through them step by step they are
simple enough. I will write out the calculation and put it on the web
as note 4, I hope that helps.
4 Nov 2002
Question about
schols. Hi, Could you tell me if all the course covered in 2e2
will be on the schols or if only the part of the course covered up to
the schols?? Is there any futher reading that you recommend?? If so
what book(s)??
Reply The schols will only examine material
covered up to that point. There is no extra reading recommended, but
it might be useful to look at previous scholarship papers. Two of
these can be found on my website.
11 Nov 2002
Question about
Stroud. You're lectures are great. I used Engineering Mathematics
by Stroud last year for 1e1 and 1e2 and found it a great help. I was
wondering if if would be any use for me this year or is it really just
for first year and hence to simple?
Reply Stroud is pretty
good book and if you have a copy I would keep it for revising first
year things we use again this year, for example. linear algebra and
inhomogenous equations. However, not much of the material we do this
year is in Stroud, there is a chapter on Laplace which it is helpful
to read, but there is only a small bit about solving differential
equations at the end, whereas most of what we did was solving
differential equations. Thanks for being nice about the lectures.
12 Nov 2002
What are Laplace transforms for? Just wondering
if you were always good and comfortable at maths,it's such a task most
of the time,what kind of things can we use this Laplace stuff for?
Reply Maths was always my best subject but I only became really
interested in it in my third year in college. Laplace transforms are
used to solve inhomogeneous differential equations of the sort we have
been looking at, differential equations of this sort are very common
in all sort of engineering, particularily electronic engineering. The
big advantage of Laplace transforms is that they provide a very
methodical way of solving differential equations, other methods have a
bigger element of getting the knack of it.
14 Nov 2002
Query about link to contorsionists page Love the pictures on the website, Don't really understand the reasoning
behind the contortionists though. Keep up the good work.
Reply This is a bit off topic, but, first, I think some of the
contorsions are pretty amazing and, second, in a lot of the pictures;
the old circus pictures, the mail-order wives style pictures, the
child gymnast pictures, there is a striking contrast between the
talent and hard work behind the image and the tawdry or exploitative
purpose to which they are being put. I found this aspect quite moving.
6 Dec 2002
Christmas message Happy Christmas,got a lot of maths to do over the hols ,hope you don't
get flooded again,pass Seaview AV, every day,can I drop in for COFFEE and buns
if it's very wet!!!And nice hair in the Rocket club,how long ago was that??
Are all particle physicists savage mathematicians as well?
Reply None really.
6 Dec 2002
Dog racing Read your intro to dogracing
eagerly, however nothing useful I can use to smash bookies, how you
accuaretly model anything like this, if the whole Irish Racing sceneis
as bent as it reportedly is?Also how would one get into setting
odds,and is risk assessment linked to this type of thing? Do
oppourtunities exist for Engineers in these fields?
Reply Again, this is off topic, but to answer your question, it
is true that dog racing has a bad reputation, but we were assuming
insiders always bet at the ring side and not at the tote. It might be
interesting to test this by measuring which odds better reflect the
true odds. The project on dog racing was a summer project to provide
fun work for a maths undergraduate. It was unrelated to my own
research work on soliton in particle theories, but it did make a good
summer project. Anthony Brew worked on the project for ten weeks, he
is still doing the odd bit when he gets time. The idea was to compile
a database of results and then buy some of these guides to betting on
dogs you see on the internet. These guides always have ad hoc formula
for chosing the best dog, you add points depending on recent placings
and on recent times and so on. Next, we planned to take the weighting
in these formula and to optimizing them against the actual results
using an optimization technique called simulated annealling. After
that, we intended to go back and see how the optimized evaluation
formulas related to the true odds and use that to come up with a way
of estimating the true odds so that these could be compared to the
bookies odds. We never got this far.
20 Jan 2002
Scholarship
exams Hello there, I am interested in sitting the Scholarship
exams next March and was hoping you could give me a list of topics
which will need to be covered for the exam, including those already
done in lectures. And also any helpful hints or tips for what I might
need to do.
Reply The scholarship exam is based on the
material we have covered up to the time of the exam, there is no
special topics. The previous scholarship papers should give some idea
of the style of questions.
22 Jan 2002
Website Why isn't
the 2E2 website being updated? I need to see the tutorial sheet
before I turn up at the tutorial or else it becomes time
wasted. Please resolve.
Reply The 2E2 website is updated
almost all the time. Putting the tutorial sheets on the web in advance
was an experiment, after discussing it with various people, the
tutors, some students, and so on, I decided it wasn't working, most
people didn't look at them in advance, some people wrote out the
answers before hand and handed them up in the tutorial. Since the
tutorial mark goes to the exam, this isn't fair. The best thing, I
think, it to give the tutorial sheet out in the class and then you can
try it there and then, if you have problems, ask the tutor. The best
way to make sure the class isn't wasted is to go over your notes
before hand. I am constantly thinking about how to run the tutorials
because they are an important part of how the class is taught, so I
would welcome any suggestions about how it could be changed for the
better.
23 Jan 2002
Tutorials changed for the better
Hey! You asked for suggesstions on making (y)our tutorials better and
more worth while... I go to your tutorial on a Wednesday morning. It
is a bit of a pain in the ass trying to get the tutor's attention,
'cause there's a good few of us in the class... So, it might be an
idea to either; have 2 (or more) tutors in the class, or get the
respective tutor to do examples on the board of commonly asked
questions ('cause the guy I go to is always explaining the same thing
over and over to; say for example 3 people, and then the next 3 people
will ask exactly the same question... or similarly, a large group will
just gather around him while he's showing one person how to do one
question). Instead of this, maybe he should just address the whole
class, and whoever is having problems can listen etc... But other wise
pretty cool course! Keep us entertained!=) Cheers!
Reply
First of all, sorry it has taken me so long to reply, I wanted to
discuss your suggestion with the other tutors and kept forgetting it
when I saw them. You have two suggestion, the first is that we start
doing the examples on the board in the tutorial class. I am against
that because it would make the tutorial just like the lecture: I try
to do lots of examples there as well. As you say, the tutor usually
ends up saying more or less the same thing again and again, but, I
think the point is that it isn't exactly the same thing, it is always
modified slightly to take into account what whoever it is has managed
so far and how they seem to be understanding what you are saying. This
happens almost sub-consiously, if the people around you are looking
confused you slow down and try to think of a better way of saying it,
if they are smiling and nodding, you speed up and go on to the next
point. This doesn't work so well if you are just talking at the board.
Also, the student sometimes has question based on what you tell them
and that allows you to learn exactly what it is that is hard for the
class. The problem is that this is no good if the tutor never gets
around to you and here the problem is that there is only so much
tutorial time, there are limit resources and limit free hours in the
time=table. It is a bit better this year than last year, there is an
extra hour and the MEMS students are taught seperately. The thing
seems to be that the extra hour is at an inconvenient time, 1 to 2 and
not many people go to this class. your other suggestion, having a
class with two tutors is one possibility, it is hard to know how this
would work in practise and I will discuss with the engineering
department the possibility of giving this a go next year on a trial
basis. One danger is that the tutors might just spend the hour
chatting to each other.
12 Feb 2003
Praise for course
i think your class is sound, its about the only course i can
understand. fair play to you.
ReplyGood, thanks.
21 Feb
2003
Wants a Laplace transform revision Any chance of doing
a quick revision lecture on lapace transforms before the easter break,
as they come up quite a bit in the 2e7 exam we have then.
Reply I can't really I am afraid, the curriculum for the
course is quite long and I don't have a spare lecture. If you want
help call to my office and I can go through it with you, or, if you
want we could organize a help class of some sort. Email me.
26
Mar 2003
Question about Q4 PS 13 Just a quick question bout
Q4 on tutorial sheet 13. When you say the exp(2t) is turned into a
redefinition of C (in the web solutions) does that explain where the -
1/4 exp(2t) goes to cos i had a bit of difficulty following that. I'm
guessing its a perfectly legal thing to do and all but would an exp
not be an important term to keep in the final answer?
Reply Yes, what you said is correct. The C is an arbitrary
constant which in pratice is set by the initial conditions. Now, in
the solutions you end up with
y=C exp(2t)+t/2+1/4-1/4 exp(2t) (1)
where C is an arbitrary constant. Lets define a new arbitrary constant called A where C=A+1/4, then the solution is
y=A exp(2t)+t/2+1/4 (2)
where A is an arbitrary constant, it is just as good an arbitrary constant as C. Say we were told that y(0)=1/2 then in equation (2) we would have, putting in t=0
y(0)=1/2=A+1/4
hence A=1/4 and the answer is
y=1/4 exp(2t)+t/2+1/4
whereas using equation (1) instead we have
y(0)=1/2=C+1/4-1/4
so C=1/2 and putting that in gives
y=1/4 exp(2t)+t/2+1/4
exactly as before. Thus, to answer your question, the 1/4 exp(2t) is absorbed by defining a new arbitrary constant.
Now, rather than go to the effort of naming a new arbitrary contant and writting down how it related to the old one, it is common just to say "by a redefinition of C we . . .". The important thing is to remember that the term with the arbitrary constant is of the form constant mutliplying exp (2t) so this trick only gets rid of terms that look like this.
3 Apr
2003
Exam questions difficult hi! your web page is
brilliant! ive looked over all the solutions you have for laplace an z
transforms and understood them ferly well but when i tried to do the
exam papers i found them very difficult???
Reply You need
to bear in mind that exam questions are set so that it is difficult to
do all the question, so remember that alot of the marks go for the
early part. if you have practised the problem sheets and have some
idea of what is going on in the questions you shouldn't do too badly.
Which exam questions did you find particularily difficult? Also,
remember that we did less Z-transforms than in previous years, there
are some past exam questions on Z-transforms you will find very
difficult, for example, we didn't do Z-tranforms with complex numbers
or with stability or signal processing. I am hoping to do a bit of
partial differential equations instead.
13 May 2003
Solitons
I know this is unrelated to 2E2, but I read on your website that you do
resaearch into BPS monopoles & skyrmions. What exactly are BPS monopoles &
skyrmions?
Reply There is a quick explaination here. The idea is that solitons are complicated solutions to nonlinear equations that seem to behave like particles. The hope is that they might explain which equations describe particle interactions because it might turn out that these equations have to allow for the solitins to exist.
13 May 2003
Some questions hello! some questions if you please:
1. what is the official definition of the laplace transform and what range of
t is necessary to define it? (question 1 , 2001)
2 is it necessary to know how to derive the heaviside function?
3. in lectures you have said that the syllabus has changed this year and that
some questions or parts of questions from previous years have not been
covered, could you please highlight what does and does not need to be
covered.(with reference to previous exam papers)
thanyou for yor time,
Reply 1.
The official definition of the laplace transforms is
L(f)=int_0^\infty e^{-st)f(t)dt
I never discussed the range of t needed to define it, I tried to stay away
from such formal matters, but implicit in this definition that you need f
for t from zero to infinity.
2.
Do you mean, do you need to know how to derive the Laplace
transfrom for the Heaviside function? Yes.
3.
I have a list of this on the web already on the past papers page.
13 May 2003
Heat equation Hi I was just wondering if there would be any chance of putting up some notes
on the material covered in the last term, especially on the heat equation.
Reply I am hoping that the solution to the last problem sheet will guide you through the heat equation.
13 May 2003
2002 paper hard Hi! I was doing the 2002 Summer 2E2 Paper, and
was alarmed at the length and difficulty of the inhomogeneous
differential eqns question (Q5) & to a lesser extent linerization
(Q6). I find it's hard enough to get one of the two parts done in the
time required, nevermind two! Will our exam be like this? Otherwise,
your course has been grand all year. Thanks!
Reply I agree
that these questions were quite difficult, but they would have been
marked accordingly, when I am marking I go through a quantity of the
papers and consider the answers and then think a bit before coming up
with a marking scheme intended to make the average mark obtainable for
each question the same.
16 May 2003
PS14 i dont understand where line 7 of question 1 on tutorial 14 comes from!
ReplyDo you mean eqn 7? So here, we write,
y=f_1x_1+f_2x_2
and then substitute it into the equation. On the left hand side the differenciation gives f_1'x_1+f_1'x_2. Now on the right hand side we know that
Ax_1=5x_1
and
Ax_2=-5x_2
this is by defination of the eigenvectors, hence
A(f_1x_1+f_2x_2)=5x_1-5x_2
and that gives eqn 7. Let me know if this hasn't answered your question,
16 May 2003
Linearizing Just a couple of quick questions; firstly, how much do our tutorials count
for in the summer exam, and is this fixed or can it be changed to give the
highest mark? (I haven't made all the tutorials).And secondly- i can't get to grips with linearization, particularly the substitution of eta. For example, at the second critcal point in tut.15 q3 i can't see how y2 dash = -eta -y2.
Apart from that, i'd just like to say thanks alot for all the time and energy
you've put into your website and lectures over the past year. It's much
appreciated and you're by far the best lecturer i've had since being at
trinity.
ReplyThe tutorials count for 10%, so if you haven't made them all you shouldn't
have lost too much. As for your question about linearlization,
well, y2'=-y2+y1(1-y1), now one of the stationary points is y1=1, so, to
be near this point we let y1=1+eta, you see, if eta is small then y1=1
plus or minus something small and hence is close to one. Substitute that
in and we get y2'=-y2+(1+eta)(1-1-eta)=-y2-eta-eta^2 but if eta is small
then eta^2 is really small and so we can approximate by ignoring eta^2,
this is always the plot, ignore squares and cubes and so on and use the
sin(theta) approx theta and end up with an approximate equation which can
be written in matrix form.
Thanks for your kind remarks about the course and good luck in your exam.
19 May 2003
Notes requested
In the past paper link you say that the Autumn 2002 paper solns. will
be available "by next wednesday" etc. Will these be available before the exam?
Also I am finding the last tutorial very difficult to follow (Heat Equation).
You mention that we did a different version in lectures as opposed to the one
on the problem sheet. Are we expected to be familiar with both for the exam? I
find going through past exam solutions are a great help in revision but we I
can only find the one heat equation question on tutorial 23. Would it be too
much to ask for you to make available one more heat equation question and post
it on the web?
Many regards,
P.S. You course was by far the best course I've had since I came to college.
Reply I am sorry about this delay, I have been delayed. The first of the autumn papers is on the web now. I have put up a modified version of PS23 with the other boundary conditions, this is note 27. I hope it will be some good.
20 May 2003
Linearization 1. is the 2000 paper worth doing as a lot of the questions seem extra
curricular? 2.with regard linearization; could you please explain the first couple of steps and im also not sure about this "eta" craic?
Reply you would be better off going over past papers than doing the 2000 paper,
the course has changed a bit and the emphasis now is more practical. As
for linearization, the first step is to rewrite as two first order
equations, let y1=y and y2=y1'=y', then y2'=y'' and you can see what y''
is from the original equation. Next, find the the stationary points, this
is where y1'=y2'=0, eg if y1'=y2 and y2'=y1^2-y1 then the stationary
points are at y2=0 and y1=0 or y1-1, now, approximate the system near each
of the saddle points, so, if the stationary point is y1=0 then let y1 be
small and approx, eg if you have y1^2-y1 then that is approximately y1 if
y1 is small. if the stationary point is not zero let y1=stationary point
plus eta, and make eta small, then if eta is small you are at the
stationary point, so, for example, if y1=1 is the stationary point, let
y1=1+eta and make eta small, then
y1^2-y1=(1+eta)^2-1-eta=1+2eta+eta^2-1-eta and this is approximately eta.
I hope this helps, the key is when approximating, drop squares etc and use the sin(theta)=theta approxiamtion,
20 May 2003
Kind remarks just like to convey my
appreciation for the work you've done especially on the webpage
putting up solns etc, stuff that you probably dont have to do to get
paid. cheers for the year you were great entertaining and still got
through the work. did you ever have aspirations to do stand up
??????
Reply Thanks a million. I also felt the lectures went
quite well this year, it can be quite enjoyable when you feel you are
generally getting on with the class, it makes me nervous about next
year, I have to admit, you know how for example if you are playing
pool and you have a good game, the next one is generally
terrible. Anyway, best of luck in your exams.
20 May 2003
Linearization
I dunno if you will get to see/answer this before the exam but here
goes ..I'm OK now with the whole linearization thing (finally!) but
how far do I have to work out a long linearization question (like a
saddle point/inward spiral one say) before a "sketch" of the phase
diagram is attempted in the exam. Is it on ce you find the
eigenvalues, and you know from them it should be a spiral and a saddle
point, or when you find eigenvectors too, or should you work out the
solu tion to the spiral & saddle point fully?? Thanks for your time!!
Reply So the main objective is to sketch it so you need to keep going long enough to get all the information you need. In the case of a spiral, you really have all the useful information once you know the eigenvalues, working out the eigenvectors doesn't tell you anything that will help you sketch. The saddle is different, knowing the eigenvectors helps you draw the saddlepoint because it tells you where the lines are going, trajectories one side if the incoming eigenvector go one way, on the other side they go the other way. For circles you need to work out enough to know what the scaling is, ie, is it an ellipse and if so, what shape. Of course, all these remarks refer to getting full marks, if you are in a hurry, you would get lots of credit, but not full marks, in the saddlepoint case without working out the eigenvectors.