The 2018 exam was released for use as a Mock exam. Solutions now available.
Live streaming revision sessions (you can replay by clicking on links below). I'll be broadcasting from my office on using my phone using periscope.tv under the handle @doctorporter. Here's a schedule of broadcasts:
Preliminary course notes, problem sheets and problems class material will be made available in advance of the lectures. Exams and exam solutions for the last three years will become available before the end of the course.
The assessment of the course will be based on the material presented in the lectures as this constitutes the formal account of the course. The notes provided below support the lectures.
There will be 5 problem sheets and homeworks will be set from weeks 2-6. Students are expected to hand in homeworks for marking and feedback though marks do not count towards the final unit mark. Solutions will be made available after homework hand in. Additional video tutorials are provided to help guide students through answers.
Any electronically prepared handouts from problems and exercise classes will appear below. Solutions are done by hand in lectures and not available electronically
I've delivered this course in the previous 4 years
Feedback is no longer active because it's the weekend
Q: When you did Q1 on PC5(sic) at the end of the livestream didn't you do it
with an anticlockwise orientation but the question states clockwise?
So wouldn't the path go all the way round the circle instead?
A: Did I ? Guilty of not reading the question !!! Tut tut me.
So I should have gone
round three-quarters of the circle in the other direction instead
as you say.
Q: Do you have any tips on figuring out the best parametrisation for a
surface? (Thanks for all the material you post on here!)
A: Only as I've said them in lectures (also see last periscope revision
session).
Q: (Thu 10th) Thank you for doing the live stream revision sessions,
I found them very helpful.
A: Thanks, and I'm glad you found them useful.
Q: (Wed 9th) Thank you so much for all of the periscope videos you've posted!!
They're so useful and you're an absolute ****
A: I'm glad you think they're useful.
Q: (Wed 9th) Is there any chance you could post the solution to q2dii on the 2016 paper using the mapping (rcos(theta),rsin(theta),4-r^2)?
A: I don't have anything more than what I've already posted.
Q: In problem sheet 4 exercise 6b, for example, how did I know that the
direction of N(r, theta) is correct??
A: You aren't given a direction in that question, you are asked to
show that Stokes' theorem is satisfied, so whatever dirn the surface
points, it must be consistent with the direction of the path integral
around the edge of S.
Q: Hi I think there might be a mistake in the Jan 2015 mark scheme -
Isn't the answer for 2)c)iii) 2*pi? Just wanted to clarify
A: I don't think you should worry about the mark scheme for a past
exam. The answer is 2*pi, I don't see the problem with this unless
you believe that 2 divided by 2 is a half, as I wrote.
Q: Could we simplify 3delta(ij) anymore? Does it equal 3?
A: Um. Oh dear. You have a definition of delta_{ij} in the notes, why not use that ? If you believed your own question, it would imply delta_{ij} = 1 and what would be the point of using something which is 1 when you already have the number 1 ???
Q: Will the formulas for the transformation of the divergence and the
curl and the Laplacian be given in an exam, or are we expected to
memorise/be able to derive them?
A: I've covered this so many times in so many different ways.
Q: how can I understand the direction of N(r,theta) in surface integrals,
since I don't have (in most exercises) a diagram? Therefore if for
example I want it to be outwards and N(r,theta) greater than 0, can I understand
something?
A: I don't understand your question. N is a vector, so asking if it is greater
than zero is missing the point. And you are always given a description
of the surface, so you can draw a sketch of that surface if you like.
Q: (Wed 9th) Perhaps if you are not going to do the 'difficult questions' today,
would it be possible for you to post online the ones you would have
done so we can give them a go? Thanks
A: It would require substantial effort on my part to put together some
tougher questions and would have no or limited benefit to almost
everyone in the class. I think we're good where we are.
Comment: (Tues 8th) 2*thanks
A: Glad it's helped.
Q: (Tues 8th) In the lecture notes, when proving for transformations of the
divergence and curl in chapter 2 you used identities not given in the
notes i.e. in 2.8.4 you used the following:
div((h1u1)*q1/h1) = grad(h1u1) x (q1/h1) + (h1u1)*div(q1/h1)
Do we need to know this in the exam and if so could we have more
insight into this?
A: I did these identities in lectures -- the lectures are the official
record of the course and the notes support the lectures. They are
also covered in questions on the problems sheets.
Q: (Tues 8th) Please can you proceed with the difficult questions video that is set
to take place tomorrow. I would find it very helpful. Thanks.
A: I appreciate your interest and respect your request. I'm not sure I
will. It's a lot of effort from me if there's low interest, which there
seems to be... Understandable, as students need to be doing revision, not
watching me doing it for them !!!
Comment: (Mon 7th) More lecturers should livestream on periscope.
A: I'm glad you think so... I think we should try and catch up with
the modern world (and I'm a luddite !).
Q: (Mon 7th) For the 2018 paper question (2biii) would there be "error
carried forward" marks if you did (2bii) incorrectly and ended up with
the wrong value for the integral?
A: Yes, markers penalise you only once, unless you're carrying forward
something which is so absurd you should stop at every later calculation.
Q: (Sun 6th) Would you be able to go through questions 9b/c from problem sheet 2 if
there's time in the session on Tuesday? Thank you.
A: Certainly. Will do. Thanks.
Q: (Sun 6th) are there more examples to the implicit function theorem other than
PS1 Q7?
A: I don't know. There's a past exam question/problems class Q I think.
Q: (Sat 5th) The definition of a path states it to be a bijective function,
yet a closed path implies that P(t1)=P(t2) on the interval [t1,t2] how is
this possible if t1 /= t2?
A: Fair enough. I thought I defined my path p(t) for t in (t1,t2).
In which case, you need limits. But hey, let's not argue about single
points in an inteerval of a smooth function when you're integrating.
Q: (Sat 5th) How important are the 'Interpretation of the gradient' (2.3.1 and
2.3.2) since they were never used in any homeworks or problem classes?
Do not know what to revise for them.
A: Point taken about lack of questions. E.g. there's a past exam with a
(difficult) question where you need to know about this. In this sense
it's just as important as everything else.
Q: (Fri 4th) I think there was an annotated version of the revision sheet covered
in the live video today, which appears to have disappeared from this
website? Was really helpful so would be great if the! link to it could
be replaced!
A: Always there, still there.
Q: On page 29 of the lecture notes I'm a bit unsure of the working with
cartesian coordinates. How does oddness of the function wrt u and v
remove the first two terms in the integral, and how do we know that
the function is odd?
A: An function odd in u and v satifies f(u,v) = -f(-u,v) = -f(u,-v).
Integrating an odd function over a symmetric domain gives zero. E.g.
integral of sin(x) from -1 to +1.
Q: can you please post the solution to problem class sheets? Thank you.
A: There aren't solutions written down (I said this in class and it says on webpage). I do
them in class.
Q: could you post answers to the problem class sheets that you didn't
cover in the lectures please? for example, for problem sheet 6, Q1,3,7
A: See above. There aren't solutions for me to post. I could do
some of Q1,3,7 on Tuesday's problems class.
Q: If we know what the vector N will be my memory do we still need to
show the calculation with the cross product of ds/du x ds/dv?
A: As a general rule: always show me in your exam solution
like I would show you in a problems class/lecture. So I
would do the calculation. Unless it's a definition or a result which
I clearly say is "quotable", do the calculation.
Comments: ~179 people (the broadcast is public, so not just MVC class presumably !!) watched the live stream problems class on Friday, and ~6 said it was useful. I don't know if that counts as success !!! I've added a direct link to the broadcast so you can watch it on replay. If you send me specific requests (some already have) about what to cover, I can do that in Tuesday's session. For now, I plan to do the 2018 exam on Monday pm and be "responsive" on Tuesday.
Q: For question 2c iii) in 2015 exam paper,the vector N points inward the
surface, why you didn't change the sign of N to make it point
outwards? However, in question 4 problem sheet 5 you did change the
sign. Thank you.
A: See comment below about 2015 exam.
Q: Does periscope record the problem classes like replay so we can watch
back or do we have to watch it live?
A: I think you can watch it on replay. If I get the option to save it
for later, I'll use it. But it's new to both me and you.
Q: If asked to state stokes' theorem or the divergence theorem can we
just write the equation or do we need to explain what all the
variables represent?
A: You need to say what the theorem says and therefore explain the
variables.
Comment: of all the modules I'm taking this year, MVC (and complex functions)
are my least favourite (big up pure maths) but you were a really
interesting lecturer. Now that I'm revising, i feel like i understand
the course way better than i expected to. Thank you for all your time
and work, I really appreciate it.
A: Thanks for the kind words. You do realise that anyone who enjoys
pure maths needs their head looking at ? Seriously, though, this
happens a lot with this course -- people don't often "get it" in the
lectures and suddenly do "get it" when they spend time revising it.
The lecturer clearly hasn't done a good job if this is the case !!
Q: on problem sheet 4a in the integral how do you get from the cos to the
sin ( I know this is a silly question =)
A: Sorry, I don't know which question you are referring to ? I need a PS#
and a Q#
Q: For question 4 problem sheet 5, should the 2nd term of N be positive
instead of negative as a result of the cross product? How would this
affect the direction of N?
A: No, I'm sure the answer is correct.
Q: Could we please have a video solution for Q4 sheet 5 please? I'm a bit
confused about the directions of the normals and how you can tell if
they point inward/outwards.
A: I did the vid solutions a few years ago and the tech I used has packed up. I'm not going to fight with it again for one question... sorry. I could do it on a live stream prob class in the Jan 7th week if you ask again !!
Q: To confirm, Is the derivation for the transformation of divergence
examinable? (I have it written down as non examinable) Also which
formulae for transformation o gradient, divergence and curl will be
provided and which will we have to remember? Thanks.
A: The general long-winded derivation of the transformation of the
divergence is non-exam. Nor the curl. There are Qs on problem
sheets (PS3, Q2/3 ?) which calculate the divergence directly.
I expect you to know the transformation of the gradient. This is
a calculations-based course, not a proof-based course so my main
aim is to test if you can make calculations, not remember long
proofs. See the past 4 years exams... this years exam will be of
the same style.
Q: In the 2015 exam, problem 2.c), why is the surface integral of curl(f)
positive, since we get that the normal N points inwards, so we should
change signs and get the negative value?
A: This is a badly written question (the first year I gave the course
and set before I'd taught it). I should have specified the direction
of the surface. So I allowed either +/- answers as long as part (d)
was consistent with the answer to part (a).
Q: For problem sheet 2 question 2ii the solution gives (b.(cxa))a however
the question asks for (a.(bxc))a . Can you please give some
explanation to how you get from what is given in the solutions to the
final answer? Thanks.
A: They are the same since a.(b x c) = b.(c x a) (in notes).
Q: Are we expected to know and use both methods for calculating the
directional derivative?
A: Yes
Comment: Really appreciate the video solutions, they are very helpful
A: Thanks.
Q: at problem sheet 3 exercise 2c there is a different answer in the
solutions from your video answer. Which one is the correct answer?
A: If you listen to the commentary on the video solution, I say something
like "I can't be bothered to finish this off properly so it will have to be
checked but I think...". And I subsequently miss a term out which is
on the written solutions. If you look up the definition of divergence
in Sph. Polars on the interweb you'll find it agrees with the written
solutions. Or you could check the maths yourself !!!
Q: For the cross product of 2 vectors why is it (u2v3-v2u3)e1 +
(u3v1-v3u1)e2 + (u1v2-v1u2)e3 instead of (u2v3-v2u3)e1 - (u3v1-v3u1)e2
+ (u1v2-v1u2)e3? Why is it a + sign instead of a - sign?
A: The first is right and the second is wrong. But (u3v1-v3u1)e2
= -(v3u1-u3v1)e2 owing to the usual rules of arithmetic.
Q: could you post the solutions for the remainder of the questions on the
problem sheets if possible? thanks. (just written solutions are fine
if videos are too time consuming)
A: All the written solutions are available to download... am I missing something ?
Q: Are all proofs required knowledge, including informal ones?
A: Everything apart from when I say "non examinable" is up for grabs. But if you look at the past exams you'll see that that style of questions are more calculation based rather than proof based.
Q: (follow up on 3 below, I presume) we have had ODEs Mock exam and I must say that they turned up more
than half of students so I think It would be great to have a mock exam
A: UPDATE. I've got a simple solution to your suggestion. I will use
the previously unpublished 2018 exam as a mock. I'll release it and
advertise to people that it's available and then release solutions a
short time afterwards. You can check to see how well you did !
Comment: enjoyed your lectures, thank you
A: Thanks !!
Q: any tips how to paremetrize when using stokes theorem, always in
examples given we have different paremetrization and this is
confusing!
A: The point is (and I said this in lectures), there isn't a set method
for doing parametrisation. The reason there's a lot of different examples
reflects this and it's my duty to cover lots of different ways of doing
it rather than sticking to a very narrow range of examples. The mantra
we have is find two variables which allow you to walk across a surface
and those are your parameters.
Q: are we going to have revision classes before the exam?
A: Difficult to see how, as there's no room in the teaching timetable.
Even then, I'm not sure what a revision class actually means. You
need to do the revision, not me !! I'll try and find a way to
provide some form of extra support for people that have questions
and problems they'd like me to go over. I could give a mock exam, but
I tried that 4 years ago and only 2 people out of 180 tried it...
Q: can you upload problem sheet 5 solutions please?
A: Yes ! Done. Sorry I was a bit late, but I didn't realise the week had gone so quickly.
Q: What is happening in terms of having a feedback class for the 4th and
5th homework sheets? Will there not be one or could the first Methods
of Complex Functions feedback class be used to go over these sheets?
A: OK. So this is out of my control. When the course is set up I get given 3 problems classes and three feedback classes. I set 5 homeworks, so there's no way you can feedback on all of the homeworks in a natural way. It's nuts, but I'm not the person in charge of designing the system. My advice: ask the feedback class leaders to use the MCF sessions to go over outstanding MVC material.
Q: I don't understand how to find a path for a given curve/surface even if I can visualise it. Could you go over some tricks or techniques to find one if we have any time in lectures?
A: I'll try my best.
Q: Will this site with all of the problem sheets and solutions stay open
for access until the January exam is done?
A: Yes, it's the course website, so it stays in play until all our business is over.
Q: is dxj/dxm = deltajm or = deltamj?
A: It doesn't matter: delta_{jm} = delta_{mj}... the identity matrix is the same as the transpose of the identity matrix.
Q: Can you please do a video tutorial for Problems set 3 question d & e?
A: Alas, my camera doesn't work anymore. You'll have to ask math-info to buy me a new one.
Q: Problem sheet 4 appears not to be working
A: I haven't released the HW sheet for this week yet.
Q: Would it be possible to write on the projector so the lectures can be re-watched on Re-play ?
A: I can see that being able to replay the lectures would be good for some people. It's a tricky one for me and you. I tried using a projector a couple of years ago and I feel very strongly that the lectures I deliver on the board are significantly better than if I use the projector. There's another aspect to this which you might not agree with. When you get a job and you have to attend a meeting with clients/bosses, you don't get a chance to replay that meeting. You may not have caught everything you needed and wish you'd been able to go over something again. But what you'll do is go away and use your initiative to fill in the gaps and come up with the best solution. And so there's this element of "real life" in lectures which I believe is healthy from a personal development perspective. That is, you have the potential to learn more by not listening to me again. Sorry if that sounds condescending. Finally, remember that this course will finish with 6 hours of video tutorials which cover most elements of the course, from a practical problem solving perspective.
This Q arrived looking like this: "Is Ýxi/Ýxj always ƒÂji". I think it's supposed to be "is d x_i/d x_j = delta_{ij}".
A: Yes.
A: After all the difficult deriving using every rule on the planet, can
you get zero for the laplacian?
A: Yes :/
Q: would it be possible to write on the projector? The blackboard is
really hard to read.
A: My handwriting is the same on the projector and the blackboard. In fact, it's worse on the projector because I can't rub out mistakes. Is there a problem with lighting in one (or more) of the lecture theatres ?
Q: Hey, on the section of when you talk about transforming the gradient,
is grad(f) in equation (20) the same as the grad(f) in equation (19)?
I think they are both the same, but one in (19) is written in terms of
x, y, z, and the other is written in terms of q_1, q_2, q_3.
A: Yes, what you think is exactly right.
Q: It would be a lot easier to use the projector instead of the
blackboard in lectures.
A: Easier in what way ?
Q: Could you please slow down the pace of the lectures. I have been a
little bit lost. Thank you.
A: OK, I do understand your concern. But (i) I need to get to the end of the course, so there's not much I can do with the pace (ii) I'm going slower than any previous year (iii) I haven't sped up, I'm just covering concepts which are a bit different and harder than before, so it feels fast.
It'll just take a bit of time, but by the exam you'll be fine with the material.
Problems with web links sorted now, thanks. Also typo on HW3 instructions fixed. Cheers to all.
Q: PLEASE improve your handwritting on blackboard!
A: Thanks for this: I will try to do better !
Q: Are there extra questions you would recommend doing from Marsden's
'Vector Calculus' to accompany the homeworks?
A: I don't have an "additional" set of exercises. Anything that is
doable is a good exercise.
Q: When explaining theorems/results or examples in depth, could you describe them more simply briefly beforehand? Just to know what we will do with it or use it for, for instance, like saying, "the implicit function theorem will let us determine whether we can find solutions near a point for these types of functions." That is oversimplified, but it hard to see what the result actually is of a
theorem or example sometimes.
A: I do try to do what you suggest, but I will redouble my efforts. Sometimes it can be hard without writing out a definition or a theorem first to set the scene coherently.
Q: Where/when will the homework solutions be published ?
A: Handouts are provided in lectures and online versions uploaded after handing in dates have passed.
Q: In lectures for Multivariable Calculus, for summations you have been
calling the symbol for it epsilon, however it is actually the Greek
letter capital sigma.
A: Really ????? Sounds like an odd thing for me to do.
Perhaps I have, in which case I apologise.