Questions, comments and answers.
11 October 2007.
Text book I see you have very helpfully put a link to useful textbooks for 231 on your webpage. Unfortunately you also left it blank. Does this mean that no useful texbooks exist?!
Reply That's weird the page shows up fine for me, I can't think why it wouldn't work for someone else. There are lots of suitable books, on the text book page I suggest are
- Erwin Kreyszig, Advanced Engineering Mathematics.
- Glyn James, Advanced Modern Engineering Mathematics
- Howard Anton, Irl Bivens, Stephen Davis, Calculus. (Contains only the first part of the course: double integrals, line integrals etc).
- Calculus: A Complete Course, R. A. Adams, 5th Edition, Addison Wesley Longman.
29 October 2007.
Frog
Hi Dr.Houghton, just thought I'd tell you: it's a rather strange thing to say, but the frog on your personal web-page seriously looks like you( especially if you stand sideways and put your tonue out a little bit)... Try it in front of the mirror, you'll be surprised! (P.S. hope you don't get offended, I may be the only person who sees it)
Reply You know, you are the first person ever to say that. I can't say if its true or not but it would be funny if it was.
6 November 2007.
Just wondering
Whats the difference between prof and dr in front of a person's name (feel free to take the mick ifthis is a trully stupid questiion)
Reply So, in the University and often outside it, Dr is used for people with a PhD, a degree you do after your primary degree if you want to go on to do research. For most subjects, psychology can be an exception, a PhD is an apprentice style research degree, you work with a supervisor on a research project for three or four years on these islands, with a few taught courses or for five or six year in the US with two years of taught courses at the start. Almost always lecturers have PhDs. Universities tend to be all slightly different but here there are three grades of lecturer: people are hired as "Lecturers below the merit bar", after a few years, as long as they are doing ok, they get promoted to "Lecturers above the merit bar", which is what I am. When you are a lecture above the merit bar, you can apply to be a "Senior Lecturer", promotion is not automatic and depends on research profile and achievement and to a lesser extend on teaching and contributing to the college. Leaving aside the retired people who still teach for us, we have three Senior Lecturer: Richard, Donal and Sinead. The next grade is Professor, Senior Lecturers can apply for professorships and again promotion to that grade depends on research achievement and, to a lesser extent on the other stuff. In the university and often outside it people who are professors write prof before their name. Maths actually has no professors in this sense: there is another type of professorship called a "chair professor" or a "name professor" or sometimes just a "big professor", these are professorships which pre-exist the person who holds them, often they have impressive names like the "University Chair of Natural Philosophy" and the "Erasmus Smith's Chair of Mathematics", people aren't promoted to these professorships, instead, when they come vacant, someone is found to fill them. Often some good person from outside the university is hired in and it is an opportunity for the university to get some leading international academic, Adrian and Samson are chair professors in this sense. Most universities have a system similar to this, some in the UK have an additional grade called "Reader" coming between Lecturer and Professor. In the US lecturers are called professors so anyone with an academic job at a university is called prof. Personally, I find the idea of titles weird and feudal so I try to avoid being called Dr.
6 November 2007.
Class Hey You said today that people said your class was too fast but i was talking among my friends and we like the pace.
Reply Thanks for letting me know, I will bear that in mind and try to find a pace that everyone is happy enough with. Email again if it starts to get too slow for you.
22 November 2007.
Correction
Hey conor, I'm a student in your 231 class. I don't know whether you want to be notified of these sort of mistakes, but in the notes I.II on three dimensions, you have left out the word cylindrical when explaining cylindrical polar coordinates. Anyway while I'm writing I might as well say the course is really cool, you give an interesting lecture all the time, and at the right speed too.
Reply Not only am I keen to hear corrections but there is a bounty of
between 50c and 2 EUR for pointing them out, your correction is worth
a euro which I therefore owe you. Thanks for this and thanks also for
your encouraging remarks.
24 November 2007.
Notes This is a BIG thank you for having not only the lecture notes up on the web, but also detailed solutions to problem sheets. It's a pleasure to do problems when the solutions are there to see the right answer. If only more lecturers were bothered, world would be a better place to live in:)
Reply Many thanks for your kind remarks. However, I do worry that the solutions and notes do have mistakes in them, so, please, why I am very grateful for positive comments, I don't mind negative one: don't hesitate if you can think of ways to make the web maerial better.
5 December 2007.
Probably a stupid question.. Dear
Dr. Houghton. In problem set three question 5, I can't understand why
dx is between 0 and root y instead of root y and 2. In problem set 1
question one dx was between log y and 1, which I hope I understand,
but when I drew pictures of both of them and tried to work it out they
seemed to contradict eachother. Apologies for the very specific and
longwinded problem... Thanks.
Reply
The answer to your question, I think, is just that you have misunderstood the description
of the region: it is enclosed on the left by x=0, on the right by
y=x^2 and on top by y=4:
********
*******
******
****
*
so that x goes from zero to the boundary at y=x^2, it x=sqrt{y}, I
think you are thinking of the region being on the right of the curved
part. Let me know if this hasn't answered your question!
10 December 2007.
Annual exam 2007 q1 soln Just trying to fully understand two dimensional integrals.. In the 231 exam 2006 final exam question could you tell me why it is ok to integrate just for x greater than zero?? Also why is dy between zero and 2 instead of zero and four? Thank you for your help.
Reply Yes, I thought I had corrected that: you are right on both counts, the diagram shows y less than two, it should be four and x should go from minus root y to root y. I will put the right solution up as soon as I can. Thanks.
27 Jan 2008.
Sleep Hey! I didn't get to see your talk recently (but heard it was very good) but just in case you don't know, you may be interested in this? http://focus.aps.org/story/v21/st1 Yours, etc. Some 231 student
Reply Thanks for that: it actually forms part of a current grievence of mine, we sent our paper on sleep to Physical Review Letters and it got past the referees only to be rejected by the editor on the ground that it wasn't physics and now they are publishing, and featuring, a sleep paper in Physical Review E!
29 Jan 2008.
Ah!! Schols!!! Just wondering if we're expected to be able to prove Stokes' theorem etc in an exam?? Thanks
Reply You would be expected to be able to prove it to the same degree of completeness and rigour I proved it in the lectures.
30 Jan 2008.
Small Typo
Just to tell you that in your proof of Stokes' theorem in the online lecture notes curlF has a k where I think there should be a j... Thanks for the website by the way, it's very handy.
Reply Corrected: many thanks!
31 Jan 2008.
integrals with infinite limits How exactly are those integrals with infinite limits defined? I don't remember them being defined in class, nor in the printed notes.
Reply How do you mean, you mean how is \int_{-\infty}^\infty dx f(x) defined? I didn't cover that: it should have been done last year, but they would be defined as infinite limits
\int_{-\infty}^\infty dx f(x)=lim_{a\rightarrow -\infty}\lim_{b\rightarrow +\infty} \int_a^b dx f(x)
where, I guess you'd have to do some messing to make sure for the sort of functions you had in mind the order you take the limit doesn't matter. Let me know if I have misunderstood the question!
4 Feb 2008.
exp Just wondering about exponential powers, and this is probably quite an obvious question, but say you have exp(2(pi)ix). Why can't you factorise it so that exp(2(pi)ix)=(exp(2(pi)i)^x = 1? Clearly, using this logic, we can say exp(x) = (exp(2(pi)i))^(x/2(pi)ix) = 1 for all x. I can only think that either
a) It has something to do with dividing by zero, as that's sort of what 2(pi)i is, and possibly you can't divide by 2n(pi)i for n integer.
b) There's something wrong with the formula exp(ab)= (exp(a))^b
or
c) You have to expand out exp(2(pi)i) before you start powering wrt x.
Sorry to bother you with this but it's been niggling at me.
Reply The simple answer is that the rule a^{bc}=(a^b)^c only applies to real numbers: you can get all sorts of nonesense if you use them with complex numbers. This is discussed in, for example, Wikipedia; the basic point is that the powers of complex numbers aren't always single valued functions.
5 February 2008
Umlaut Hey Conor, I was just wondering why you put an umlaut over the second 'o' in the word 'coordinate' in your notes.
Reply Yes, sorry about that, it is an american thing, american newpapers use an umlaut in a double vowel if it is not a diphthong, so, since the double oo is pronounced like two o's rather than a long o, it is umlauted. Since I have such naturally poor spelling I tend to pick up spellings in an inconsistient way if they make sense to me, so, even though I only lived in the US for a year I picked up some of american spelling and I know it is a mess. I also have the silly but unshakable habit of pronouncing z as both zey and zed, even in the same equation.
9 Febuary 2008
keyboard notation for maths Where can I find online resources for learning that keyboard notation for writing Maths - eg. \int_{-\infty}^\infty dx f(x) - that you use to answer questions on this site? I've seen it on Maths/Physics forums elsewhere too but I don't understand it.
Reply It is LaTeX (pronounced laytech); it is the standard mark up language for preparing mathematical documents, it is what we use to write our papers, etc: the problem sheets and so on for 231 are prepared using it, I write the mathematics in LaTeX and then use the LaTeX programme to convert it into typeset mathematics, since it is so common, people use it as a shorthand for typing mathematics even when it isn't going to be converted. You can read about it on David Wilkin's own LaTeX site:LaTeX Primer.
9 Febuary 2008
functions and limits of integration
Is the integral of a function defined over a closed interval even when the function itself is not defined at the endpoints of the interval? For example, when we found the fourier coefficients of the block-wave function, we computed integrals over intervals at whose endpoints the function was not defined. PS. To the best of my recollection, we did not cover improper integrals - those with +-infinity as limits - last year.
Reply The thing is it doesn't make any difference: if you go back to the definition of an integral in terms of Riemann sums, changing the integrand at individual points, as for example if you leave in or out the end points. It works like this, imagine a little box corresponding to that individual point, the narrower the box the smaller the contribution the value of that point will make to the sum, and hence when taking the infimum of the upper Riemann sum it does not contribute. It might be easier to discuss this at the board if you wanted to ask me after class. To be pedantic the improper integral is the formal object you get by integrating without limits, the integral with infinite limits is defined as the limit of integrals with finite limits, which are in turn defined using Riemann sums. I am sorry this was never explained to you, this, again, is probably because of the changes involved in getting rid of 131! Let me know if there is a particular aspect that is confusing you.
10 February 2008
Problem Set 10 Q3 (b)
In your derivation of the formula, you assume that f(0)= 0, but f(x) as stated on the problem sheet is undefined at x=0. Is there a mistake in the definition of f(x) or am I missing something here?
Reply I probably wasn't so careful about this: the fact is that integrals are not effected if you change the value of an integrand at a single point and, as we noted before, Fourier series and integrals average across discontinuities, giving (lim_{x\rightarrow a-}f(x)+lim_{x\rightarrow a+}f(x))/2 at a point a, so the answer is not changed by changing the value of $a$. I should, however, always define discontinuous functions at discontinuities, it is a bad habit not to.
11 February 2008
Problem Set I am in your 231 class. I was just on your website and was wondering was there a tutorial sheet for this week? Thanks.
Reply I know I know I realised on Saturday I had completely forgotten: I will set it today and to the tutorial next Monday. Sorry!
15 February 2008
2007 fs q6 It's probably a silly question to ask, but in 2007 Schol paper Q.6. you have f(x)= a function of t,we're asked to write the triangular pulse as a Fourier transform. Is that ment to be f(t)=...? Otherwise I don't know how to proceed with the 2 variables.
But if it is f(t), the Fourier transform is really messy and I can't see a way to simplify it... maybe there's a trick to use?
Thank you,
Reply It is f(t), I have corrected the paper on the web and posted a solution, basically it is slightly messy but you do it by integrating by parts.
17 February 2008
231.II.1 typo Just to let you know, there is a typo on page 6 of 231.II.1. There are too many t's in the word "convenient".
Reply Thanks - I've fixed it, and a few other misspellings.
20 February 2008
Time Hi, this is completely unrelated
to anything, but what would happen if the Gravitational force were
suddenly to become repulsive? Would there be symmetry with the past or
not? Thanks.
Reply I honestly don't know, I do know that
gravity doesn't provide the only arrow of time, another is provided
by the electro-weak decay of the kaon so at the very least you would
also have to change all matter into anti-matter. There are also entropy
and our conscious perception of time, but I don't really know how to
relate these arrows of time to the underlying physics. It is one of those
topics which has always confused me. We, as in the Hamilton Maths
Institute organized a series of talks about time a few years ago and
these, while interesting, certainly failed to answer your
question. The wikipedia article is quite good: Arrow of time
and I will ask my collegues next time there is a lull in the
conversation at lunch and will get back to you if they say anything
illuminating. (They didn't, sorry)
26 March 2008
Maths v TP Should I stick with my theoretical physics degree, or transfer to maths?
Reply That would really depend on your preferences and ambitions; I would need more details before I could give an advise, but feel free to contact me if you would like to discuss this.
26 March 2008
Last year and 2007 q1
Hello, I'm just emailing to ask what the changes have been to 231 this year. Could you tell me what is or isn't on for the schols this year, compared to last year? Thanks very much,
Reply
It should be the exactly same; I can't remember if series solutions where on
for schol last year, they aren't this year.
Further contact Thanks a mill, by the way I noticed on
annual 2007 q1 that (I think) at the very end of Q1a) it should be 1/3 rather than 0.
Further reply You are right, but I am confused, I thought I had corrected that a few months ago and posted a texed version instead! Where has it gone?
And later
Ok; I really think I have corrected it now again!
29 March 2008
Schols Question Eight Someone else might have mentioned this to you already, but yesterday you changed the +1 in question 8 on about half of everyone's papers to a +y; the problem with this (apart from the fact not everyone was told) is that the question is actually solvable as it was originally stated (the 1 is part of the inhomogeneous case), while the changed version doesn't appear to have a straightforward solution (I looked at it in Mathematica, the real root of the auxilliary equation is 1-2^{1/3}, and the full solution to the ODE is incredibly long and incomprehensible). I wasn't given the changed version and was able to do the question, but anyone I talked to who was told to change it ended up not being able to do it. Just to let you know...
Reply I will take this into account when marking, I am sorry. It should of been -y. Please alert me whenever you think there is a mistake, espessially in an exam paper!
14 April 2008
P.S. 18 I'm not, as such, looking for answers or tips on the homework, but I'm a bit confused by q.3 on problem sheet 18. In part (a) the "solution" that you give us doesn't appear to be a solution to the equation. (i.e. u''(x)+6u'(x)+5u(x)=/=0... Which is unnerving. I'm also a little confused wrt part (b). I'm guessing that you want us to do question three using the v''+(p+log(u'))v'=0 equation, but for u=1 any other solution of the equation will suffice as an answer (y=1*v(x) for any solution v). So the best way to go about it appears to solve it without any of this fancy y=uv stuff, seeing as how to solve it with the v'' equation involves getting the log of 0, which could take a while... Am I horribly misunderstanding something?
Reply Thanks for letting me know. I found another error you were careful not to mention, I did question 2 in class today! Anyway, you are right about q3a), but say e^{-x} is one solution, to solve the problem use the y=uv trick to find the other. Q3b) is correct, though I agree it isn't necessarily a good question, the fact y=1 is a solution indicates that the differential equation is already a first order equation for y'.
23 April 2008
problems I was just wondering if you were going to post any more homeworks, and would it be possible to post the solutions to the last 3 problem sheets? Thanks
Reply There is no problem sheet this week, but there will next week and the week after. The solutions to 16, 17 and 18 will be posted soon, maybe by tomorrow evening, I was doing the schol paper first and that is now done. If you need them urgently, the sheets were similar to last years, with slightly different numbering, so you could look for them there.
30 April 2008
Exam solutions Hello, just wondering if solutions will be posted for the remaining Qs from last year's exam paper? Thanks
Reply Yes, I can do that, I will get them up over the next few days.
31 April 2008
Solution to 1a, 2007 Hi, is the solution to Q1a for the 2007 paper correct? When integrating wrt x, should it be x^5/15-x^8/24? The solution says the second term is x^7/24? Might seem silly but I just want to make sure!
Reply Yes, thanks, you are right, I will change it now. I think the next line is correct though.
5 May 2008
limits of integration Hi Conor, is there any good way of going about finding the limits of integration? I'm having some trouble figuring out where you get them from, particularly in Qs where there is a change of variable. An example of where I need help would be Q2 in problem sheet 2. Thanks.
Reply Sorry for the delay in replying; the answer is, I am afraid, that you just have to think it out. For example, in Q2 PS2, you start of with an integral from minus infinity to infinity in both the x and y direction, this is the whole plane, after you change to polar coordinates, you still want to cover the whole plane, so you want to go from the center, r=0, to infinity, r=infinity, in every direction, so theta has to go all the way round, from zero to 2\pi. Similarily, in Q3 PS3 the cartioid is defined by r=1+cos\theta, thus, it makes the shape of the cartoid as theta goes all the way around; r=1+cos\theta is the outside and r starts at zero, so the r limits are zero to 1+cos\theta. I amn't sure how helpful that is, if you want to come and discuss examples with me, do email to arrange a time.
5 May 2008
<3 231 is the 231 t shirt a play onsomething
Reply Not really, the heart t-shirt is a t-shirt classic, made famous I guess by John Lennon and my I heart 231 t-shirt is supposed to be part of that tradition.
8 May 2008
General email from me to the class For convenience, because a number of people are changing late into
231, the final exam mark with be max(E,.9E+C) where E is your exam
mark, out of 100 and C is your continual assessment mark, out of 10.
For those people who told me they were changing in to 231 earlier in
the year and who did some portion of the continual assessment, C will
be calculated using those continual assessments that were given out
after they changed. Please let me know if this is confusing, and, again, please contact
me, by email, using the anonymous feedback or in person if you are
having difficulty with some part of the course.
8 May 2008
Bon Voyage!
Hey, completely unnecessary (as you already said) but thanks for teaching us, twas a pleasure! Also, the effort to latex up all those notes and solutions was quite hefty, again thanks. Anyways, good luck with the biologists.
P.S. A final off-topic note, no compulsion to pay any heed; over the years, have you noticed a more than ordinary difference in teaching the maths/tp class as compared to that of engineers and scientists? Specifically, whilst we are a smaller class, I sometimes thought that perhaps we appeared as a crowd of cynics to some of your asides which I'm sure would have got accolades from other classes. Are we a tough crowd? :-P And if so, what did you make of that?
Sorry to bring focus to your asides, rather than course content but I guess you could take it as a compliment as to the material being straightforward enough usually and also a testament to your approachability (oh Mr. President!) to be asked such odd questions? Also, perhaps the difference is only some sort of parallax and not apparent at all from your POV.
Anyways enough of that, all the best.
Yours etc,
Some 231 Student
Reply So to try to answer your question, which I find interesting because I worry quite a bit about teaching: I have a different attitude to service classes as I do to maths/TP but not in the way you suggest. I enjoyed teaching engineers a lot, I like engineers because of their ethos of pragmatism, because they, as a class, are often quite warm and because they are sort of unruly, so teaching them is a challenge at a performance level, but, when I am teaching them I feel more aware of the lecture as a calculated piece of paedogogy, I teach with very specific learning goals in mind, and I use anecdotes and asides carefully as a sort of Brechtian device to establish a particular sort of communication with the class. With 231, a home crowd as it were, I feel more like I am explaining rather than teaching and I introduce asides because I have thought of something I feel you will ultimately find interesting or amusing, though, of course, I don't always get that right. Again, as it is a home crowd I always imagine that if I find something intersting or funny, so will the class. I do consiously try to break the hour but beyond that I am less calculating, less cynical when I am teaching 231. Is that an answer?
I have only taught scientist for one term, so I am looking forward to trying it out again, particularily since my own work is now close to biology.
On an unrelated note, I was distracted by q3 and runngin over, but in wrapping up today I meant to pay tribute to my assistent Jessica who has helped with 231 over the last two years.
30 May 2008
Schol solutions Hi, for Q1 of this year's schol paper - give the two 'z=...'s, how do you know which one is the upper limit and which one is the lower limit?
Reply You just have to stare at it and decide; obviously it should contain a volume and if you look, the first one is an upwards cup, the second, a downwards one. Plotting for x=0 might help, but this is part of the awkwardness of these integral questions, you have to kind of figure out how the limits fit together.
31 May 2008
PS17 Hi, I realise you might not see this
before Tuesday but, in Q1 of problem sheet 17 - how do you use the
generating function to get the hermite polynomials? I can't decipher
the online solution for it and it's missing from my class
notes. Thanks.
Reply I am alarmed to think you think I
don't work at weekends and on bank holidays! With a generating
function, you have a extra variable, h in this case, and the idea is
to expand the function as a series and then the coefficients of the
extra variable will be functions and these are the functions you
want. Here the generating function is exp(2xh-h^2) and the Hermite
polynomials are going to be functions of x: we are told
exp(2xh-h^2)=sum\frac{h^n}{n!}H_n(x)
so when you expand out the
left hand side and gather all the same power of h together, its
coefficient, up to some scaling factor (1/n!) is the Hermite
polynomial. The actual expansion is quite tedious, first, expand the
exponential using the usual formula
exp(A)=1+A+\frac{1}{2!}A^2+\frac{1}{3!}A^3+. . .
so
exp(2xh-h^2)=1+2xh-h^2+(2xh-h^2)^2/2+(2xh-h^2)^3/6+. . . (*)
and then expand out the bits in brackets
exp(2xh-h^2)=1+2xh-h^2+2x^2h^2-2xh^3+h^4/2+4x^3h^3/3-2x^2h^4+xh^5-h^6/6+. . .(**)
now, the next term in the expansion of the exponential, marked (*), includes (2xh-h^2)^4 and if you expand that out the lowest power of h is h^4, hence, (**) has all the powers of h up to h^3 fully, but it doesn't have all the h^4 terms, so
exp(2xh-h^2)=1+2xh-h^2+2x^2h^2-2xh^3+4x^3h^3/3+O(h^4
and grouping it together
exp(2xh-h^2)=1+2xh+(2x^2-1)h^2+(4x^3-6x)h^3/3+. . .
so the coefficient of h^0 is 1, of h is 2x, of h^2 is (2x^2-1) and of h^3 (4x^3-6x)/3 and so we can read off the Hermite polynmials, for example, from the sum formula the coefficient of h^3 should be H_3(x)/6, hence
H_3(x)=8x^3-12x
So, all you do is you expand the generating functional and then group all the same powers of h together, the coefficient should then be the polynomial. The thing to look out for is that you go far enough along the expansion to get all the terms you need, here we were looking for H_3 so we went to the cube term in the expansion, if we had wanted H_5 we would have needed two more terms in (*). Let me know if you are still confused.
Further question That was very helpful, thanks! I see where the answers are coming from now. I'm still a little bit confused though - given
exp(2xh-h^2)=1+2xh+(2x^2-1)h^2+(4x^3-6x)h^3/3+. . .,
how do you go from there to say that the coefficient of h^3 is (12^3-2x)/3? Or was that just a typing error? To compute the answer you have h^3=H_3(x)/6 ((4x^3-6x)/3=H_3(x)/3!). How do you know this from the sum formula?
Reply I think you have a mistake in your expansion, (*) above is
exp(2xh-h^2)=1+2xh-h^2+(2xh-h^2)^2/2+(2xh-h^2)^3/6+. . . (*)
you can't take all the h's out of the bracket in one go, it is 2xh-h^2 inside the bracket. You need to expand out the bracket before gathering your powers of h together. In the case of h^3 one term is coming from the cross-term of the square in (*) and the other from the first term in the cube. Now you compare the series you have to the formula
exp(2xh-h^2)=H_0+H_1h+H_2h^2/2+H_3h^3/6+ . . .
and
exp(2xh-h^2)=1+2xh+(2x^2-1)h^2+(4x^3-6x)h^3/3+. . .
and you match the coefficients of the powers of h.
Further question Hi, I think you might have misunderstood my question, in your initial reply you said "exp(2xh-h^2)=1+2xh+(2x^2-1)h^2+(4x^3-6x)h^3/3+. . .
so the coefficient of h^0 is 1, of h is 2x, of h^2 is (2x^2-1) and of h^3 (12x^3-2x)/3", I was just wondering where the (12x^3-2x)/3 came from?
Reply Oh, sorry, that was a typo, correct now.
1 Jun 2008
Q7(c) 2007 Hi Conor
Your solutions to 7(c) from last year's annual exam are missing the initial value part of the problem. Thought you'd like to know.
Reply Well so it is, I certainly wouldn't have got full marks. I get C_1=1 and C_2=1.
1 Jun 2008
PS13
Hi, in Q1(a), I don't understand how you get from the 2nd line (when working out f(k)) to the 3rd - does exp^ipi/2 just go to i? And the same for exp^-ipi/2?
Reply Yes: exp(i\pi/2)=cos(\pi/2)+isin(\pi/2)=i, exp(-i\pi/2) goes to -i for the same reason.
1 Jun 2008
PS7 Q2a
Would you be able to tell me why in problem sheet 6 question 2c F dot
product with the derivative of r with respect to time is e to the -t?
Reply
Well, F=lambda(x,y) and on the curve (x,y)=(e^t s, e^t c) with s and c
meaning sin t and cos t; now
\frac{{\bf r}}{dt}=(e^t(s+c),e^t(c-s))
and hence
(x,y).\frac{{\bf r}}{dt}=e^{2t}(s^2+sc+c^2-sc)=e^{2t}
on the other hand on the curve, where again x=e^ts and y=e^tc we have
lambda=e^{-3t} giving the answer. The are sign errors in eqn 49 which
I will correct now; but they don't affect the rest of the answer.
Hope that helps.
1 Jun 2008
Fourier formula I was just wondering if you will be giving the Fourier series formula with the exam paper this year or if we need to know them.
Reply There will be a formula sheet, the same as in previous years.
1 Jun 2008
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11 Jun 2008
odes and fourier series
Hi, in problem sheet 15, in Qs 2 and 3 - how are you calculating the C_ns? Are you using the normal formula for C_n? If so, I still don't really understand how you're getting them, could you please explain if possible? Thanks.
Reply I amn't sure which C_n's you mean; do you mean the fourier coefficients, there are calculated the usual way, I don't do the calculation there since it has been done before in the Fourier section of the course, the functions are all functions that have come up before. To calculate the C's in the solution to the equation I substitute y=Ce^{inx} in to the ode, so in the case of 15 Q2, I get (-n^2+3i+2n)Ce^{inx} on the left and \frac{sin an}{\pi n}e^{inx} on the right, so I solve to get C. Let me know if this hasn't answered your question.
Further Hi again, yeah I mean the fourier coefficients sorry. Do you use the normal C_n formula then? I know it's been done previously but, could you go through how you calculate f(x) in 3(a) if possible? Thank you.
Reply Ok so
2\piC_n=\int_{-\pi}^\pi f(x)e^{-inx}dx
and using f(x)=-1 for -pi to 0 and 1 from 0 to \pi, we get
2\piC_n=-\int_{-\pi}^0 e^{-inx}dx+\int_0^\pi e^{-inx}dx
and we can do the two integrals the usual way, one quick way might be to do a change of variables in the first integral x'=-x and then rename x' back to x
2\piC_n=-\int_{0}^\pi e^{inx}dx+\int_0^\pi e^{-inx}dx
amd putting it together
2\piC_n=-2i\int_0^\pi sin{nx}=\frac{2i}{n}[cos(0)-cos(npi)]
and the thing with cos's is zero for n even and 2 for n odd, giving
C_n=\frac{2i}{n\pi} or 0
2006 paper solutions
Hi, I tried downloading something to open with the pnm format, it won't work. Is there any the solutions to Q1 can be put in pdf format? Thanks.
Reply Evince and gimp certainly work; but anyway, I have added pdf
2007 annual solutions
Hello, in the solutions to Q4 - for grad_divF, why do you only use del/delx for the grad part. I didn't really know how to phrase this question... it makes sense if you see the solution! Thanks.
Reply That is just the one component, if you look there is a subscript one on the brackets, I am working out the x-component of the grad and showing it works for that, since no direction is prefered, it will then work for all three components.
2008 Schol exam
Just to let you know, on your website the link to 2008 schol exam, when you open it, it shows up as 2007 schol exam. I don't think it is the 2007 exam, just a typing error :)
Reply Yes, that's the 2008 paper, thanks, I will correct the header now.
2007 Q8 solution
Hi, the pdf document for this is not working. Downloaded an application to read source documents but if failed. Any other suggestions? Thanks.
Reply The pdf should be working now; the source is LaTeX.
Fourier formula
hi, I was just wondering if you will be giving the formula for Fourier Series again this year, or if we need to know them,
Reply Yes; see 1 Jun question above
Q3 ann 2005
Hi, is the solutions to Q3 in the 2006 annual exam - you bring pi up in front of b_n? Why is this? Thanks.
Reply No reason, it is just the pi from the right hand side, I multiplied it across to keep it out of the way but I really need not have.
Q1 fs 2008
I was just looking at your solution to the 2008 Schol exam and am confused about how you managed to integrate the first question.
If you consider the integral to be worked out in 4 steps (ie. there are 4 lines on your solution), how did you go from step 2 to 3 and 3 to 4?
Reply And well you might ask, there ia a typos. I have 6-12x^2-6y^2 and I integrate with respect to y, giving 6y-12x^2y-2y^3=6y(1-2x^2)-2y^3 and then I put in the limits, which are +/- \sqrt{1-2x^2) giving 12\sqrt{1-2x^2}(1-2x^2)-4(\sqrt{1-2x^2})^3=8(1-2x^2)^{3/2}, I seem to have lost that 8 although it is on the next line, I will correct the notes now. Next, put x=\sin{\theta}/\sqrt{2} and you get cos^3{\theta} from the integrand and another \cos{\theta}/\sqrt{2} from the dx.
Q4 fs 2008
Hi, in Q4 - how are you getting the 3rd component for your answer to curl[k x grad(1/r)]? I get a similar answer but in terms of y and x, not z.
Reply You are probably getting the same answer, for some reason I thought it looked neater if I substituted x^2+y^2=r^2-z^2 at the end.
Q4c fs 2008 You seem to be missing the solution for 4(c) on this years schol paper.
Reply Gosh, yes, put it in now, actually the easiest part of that question.
Exam Q8 I sat the 231 exam today and think there may have been an error in one of the questions. In quesion 8, Hermite's equation is given with an n instead of an alpha. This caused problems as the solution given for y had an x to the power of n in it. I didn't realise that this was a mistake in the exam and couldn't get the equation to work. However, others in the class recognised this and said they used the correct form of the equation. I'm really sorry if this isn't a mistake but i just wondered, if it was, would it be taken into account in the marking?
Reply It wasn't a mistake; we are, of course, free to call that constant
anything we like, alpha, n, or whatever. I called it n as a clue to
the answer to part b: there are polynomial solutions when n is an
integer. The n inside the sum of the series solution is a dummy index,
a different n, and can also be renamed, in a case like this the wise
course would have been to rename the dummy index to something else,
part of trick of knowing how to do series solutions is knowing how to
treat dummy indices. Of course, this was a subtle point and may have
caused confusion, that would be taken into account in the marking and
I am sure you did better than you might fear. It is sometimes hard in
advance to know what people will find hard and what they will find
easy.
Exam Q8
I am a student in your 231 class and yesterday in the exam,. You had '2ny' written instead of '2 alpha y' which was in your notes. As a result of this, I got very confused and brought 'n' into the summation so I didn't get a recursion relation at all since my a_n terms canceled resulting in my a_n+1 term equaling zero. Because of this, I couldn't do part (b) or part (c) of the question. I know that if that 'n' had been written as an alpha, I would have been able to do the entire question correctly and because of this I feel quite hard-done by. I am really disappointed because I think that this has significantly reduced my grade.
Reply I am sorry you found this confusing; you are not the only one, if you
look above I have answered another email about this and
I have spoken to someone in my office. However, there is no rule
linking a particular constant name to a particular equation and n is a
good name for this constant if it is an integer, as here, when you are
asked about the case with polynomial solutions. A dummy index can have
any name. I designed this question to test this important point.
However, as I mentioned above, it is sometimes hard to
gauge how tricky a question is before you see how it is answered and
in this case in which it seems there was some difficulty, it is likely
that the marking scheme will be designed to ensure students who showed
that they understood series solution will get a reasonable mark. It is
unlikely that you did as badly in this question as you may fear.
Delta function formula (Commented in the exam) I expected the delta function formula to be supplied.
Reply The only formula I promised, which had been supplied, where the Fourier formula. Luckily you seem to know the delta function formula anyway.
Infinite amounts of information
While this is completely unrelated to the 231 course, which is mostly finished at this stage, I was hoping you could help me with something that's been bothering me for a while? I've never taken a computing or informatics course, but is there any good reason why an infinite amount of information can't be stored in a finite number of bytes? I'm assured by C.S. students that it can't. Indeed, I believe them. I just can't see intuitively why?
Rply Well I amn't sure what you mean by infinite amounts of information, but generally, this would mean enough information to identify any element in an infinite set, that is, by storing the information, you are mapping an infinite set invertibly, to a finite set of bits, but, in fact, an infinite set has no invertible mapping to a finite set since if set A maps invertibly into set B, then the cardinality of A is less than the cardinality of B. I don't really know if that answers your question, so do email again to explain!
Further message You indeed formulate and answer my question entirely to my satisfaction. Thanks a lot :)
30 August 2008
why Charlie Chaplin?
Reply Charlie Chaplin is an appealing figure because his art's ultimate source is so clearly found in his own personal vunerability; also, it is easy to find images of Charlie Chaplin which are in the public domain. Furthermore, I had previously used Buster Keaton, so there is a tradition of using black-and-white icons of comedy for 231, why is another question.