The formal unit description for the course can be found by clicking here. Better information found on the first few pages of the notes.
Q: Is ∇ vector property, i.e. should we underline it?
A: Some people do, to emphasise its vector-ish-ness. Most
people don't bother because there is no scalar equivalent
in constrast with vector(r) and scalar(r) for e.g. So I don't
mind.
Q: (orientation example) and I originally obtained N to be pointing
outwards i.e. having positive components because I calculated N as the
cross product of ds/dphi and ds/dtheta, rather than the other way
round, which would have produced the N that points inwards.
A: Yes, that's it. You've got it.
Q: [correction about 5 comments below to 2014 paper Q3bii]
(orientation example) oh I think i get it now, what I said in my
previous question was wrong right? Basically we calculate N and check
which direction it goes in and using the right hand thumb rule we
obtain the orientation of the boundary of the surface C. In the 2014
paper Q3 part ii example we changed the sign of the components of N
because N was originally pointing out of the sphere and according to
the RHT rule the orientation of the boundary of S should have been
anticlockwise, but in the question it was given as clockwise so we
change N so that it points inwards, right? (thanks for the help)
A: Yes, I think you've got it.
Q: What is the inverse of a jacobian matrix, in the 2013, 1c) past
paper, the solution appears different depending on which line you look
at.
A: I don't understand your question. The exam Q does not ask for an
inverse Jacobian matrix. You just follow the Implicit Fn Thm.
Q: Is the proof of Gauss' Divergence theorem examinable?
A: I didn't go through it in the lectures; it is not
Q: Will the exam contain 4 25 mark questions?
A: No
Q: Okay, so in the 2013 paper Q3, part ii (using stokes theorem), the
orientation of S is inwards by this do you mean the following:
In this example N has components that are positive, so n hat will
point out of the sphere. So using the right hand rule, the thumb of
your right hand will go in the direction of n hat (out of the sphere
in this case) and curling your fingers in the direction of the
boundary of S indicates that the curve C (the boundary of S) is
orientated clockwise (since it goes west). As a result the two
orientations don’t match (i.e. n hat is positively orientated whilst
the curve C is negatively orientated as we take the anticlockwise
sense to be a positive orientation). So we have to negate the
components of n hat so that it points into the surface. So the surface
is orientated inwards?
A: The direction N points depends on how you choose your parametrisation.
That is, N is the cross product of s_u with s_v and if you happened to
have decided that u should switch places with v (which you could do
for instance in polars by deciding you wanted to write s(theta,r) instead
of s(r,theta)) then N would switch signs. So the ambiguity in the
direction of N is evident not only on physical grounds but also mathematical
grounds. Whatever... the 2013 part (b) question concerns a sphere but
is NOT to do with Stokes' theorem and does NOT require the RH thumb rule.
However, because there it is a
surface integral, the sign of normal to the surface MUST be specified
in order to uniquely define the value of the surface integral. That is,
if dS pointed in the opposite direction the value of the integral would
be negated. On 2013 part (c), the question IS about Stokes' theorem
and this involves a CONE. According to the solutions, we end up with
N pointing inwards (but see above: a different ordering of the
parameters would have N pointing outwards). Now we have to choose the
direction of the path C in Stokes' theorem to be commensurate with this
direction of N by using the RH thumb rule ... if you point your RH thumb
inwards near the top of an ice cream cone your fingers will direct you
along the top of the cone in an anticlockwise direction. Had N pointed
outwards, the RH thumb rule would have pointed you in the clockwise
direction.
Q: Calc 2 - '12. Question 3c iii) why does the normal to S always point
along r? Thanks
A: I can't find which question you are referring to. Is it on a
problem class sheet ? Sounds like you have a spherical surface in
which case the normal is in the direction of the vector from the
origin to the surface, which is, by definition, the vector r.
Q: Could you just explain how we use the right hand rule to find the
orientation of a surface S, as I often get the wrong direction of N.
A: It's the RH thumb rule and it applies to Stokes' theorem. A surface
can have two possible normals +/- N pointing in opposite directions.
In order to apply Stokes' theorem, the orientation of the boundary
curve C (which again can point in two different directions +/- C
going clockwise/anticlockwise around the boundary) and of the normal
N must be chosen so that when you wrap your fingers in the direction
of C your thumb points in the direction of N
Q: For the Green's theorem formula, why is the A1 term first negative and
then positive?
A: I think you mean Green's theorem in the plane ? If you take
curl of (A_1(x,y), A_2(x,y), 0) and susbitute into Stokes' theorem
that's what you get (i.e. there are minus signs from taking curl).
Q: On the 2013 paper, Q2 a shouldn’t the curl of u be –y in the x hat
direction ?
A: Looks like it. I went through this question in problems class week 4
so I would have pointed it out then. This was not my exam; it belonged
to a previous lecturer.
Q: On the week 5 notes, the example given immediately after Stokes'
theorem (part i), how do you make the 2w1 and 2w2 terms disappear
before the final equality?
A: This uses the oddness of the integrand (the integral of an
odd function from -X to +X is zero)
Q: on problem sheet 4, why do you go from (0,0) to (0,l) first instead of
from (0,0) to (l,0)? the question says oriented anticlockwise?
A: The question describes the four vertices of C (in no particular
order) and then tells you which way the curve should be oriented.
This is fine.
Q: On homework sheet 4 q1 the integral goes from a*rt(2-2cost) to
2a*rt(4sin^2(t/2))? I don't understand where the 2 at the front comes
from? Is it not meant to be a*rt(4sin^2(t/2)) or 2a*(sin(t/2))
A: There is a typo in the solutions ... after the second equals it
should be a not 2a ... but the 4a is correct after the next equals
and the 8a is correct overall.
Q: In week 5 of the lecture notes, section 3.3 in the hemisphere example,
where it says s = (u,v, sqrt(R^2-u^2-v^2)) how have you
got the third term in s?
A: I think this is already answered below. The parametrisation is
u = x, v = y and since the surface is x^2 + y^2 + z^2 = R^2 this
gives z = sqrt(R^2-x^2 - y^2) and this gives the final entry in
the vector s
Q: I'm confused about homework sheet 6 question 3. After you've done the
cross product etc and end up with z dr d theta, what is z in this
case?
A: There isn't a HW sheet 6. I can't see which Q you are referring to
Q: Problem Sheet 2 Q5b. How does the 1/2 arise in the second line of the
solution? Thanks in advance.
A: Work backwards: d/dx( 1/2 u^2) = u du/dx
Q: In order to help remember the cross products of the local basis
vectors for spherical polars, is it correct in visualising the 3d axis
with r in the x direction, theta in the y direction and phi in the z
direction, and then (since the basis vectors lie on their respective
axis?) applying the right hand rule on the system gives ie r hat x
thetahat =phi hat and r hat x phi hat= -theta hat etc., and with
cylindrical polars have the z in the z direction instead of the phi?
A: Um. This sounds complicated. You can always get the direction
of a cross product using i cross j equals k (in old money). This
is the same as the RH rule.
Q: In week 3's online lecture notes, you state that (on page18) that when
you differentiate r by x'(1) you get (R(11),R(12),R(13)), but in my
lecture notes from class you have summed them to
get(R(11)+R(12)+R(13)) which is correct?
A: You should be able to work it out from the definition of r. Or another
answer is how could it possibly be a scalar (e.g. a sum) when r is a
vector ? But the answer you probably want is that it's the vector.
Q: On problem sheet 3 Q2e) in the answer for the expression of the
Laplacian of f, why can’t we use the product rule to simplify the
first and second terms further?
A: You could and then your Laplacian would include 5 terms instead of 3.
It depends on your definition of simplification.
Q: Are we expected to remember any of the identities and formulas derived
in the homework like the ones in problem sheet 2?
A: I don't expect people to remember lots of things. I want people to
learn how to find answers. Learn the basic principles and practice
how to apply them to examples.
Q: On page 15, under the Laplacian section the final result has a ‘f’ in
it, why is the f there - (surely the final result should just be the
term in the brackets?)
A: Nothing is wrong in the notes. Read carefully. The Laplacian of
a scalar field f and a vector field v is defined. The Laplacian
operator is implied by these definitions.
Q: Also in the lecture notes for week 2, section 2.3.3 Eg1 , shouldn't
the grad of T at (x,y,z)=(1,1,1) be (0,-(pi/2)*e, 0). Hence the
direction in which the temperature gets hottest fastest in is [...]
= (0,1,0) ?
A: How odd that this mistake has only just been spotted. I can't imagine I
made this mistake in the lectures. Anyway, thanks.
Q: In Q3c) on the 2013 paper, is the vector field meant to be
v(x,y,z)=(z^2, xz, y^2)?
A: Yes. Dunno what happened there. It's also on Prob Class Sheet 6 where
it's correct.
Q: In section 1.6 derivatives of operations on maps, in point 3 for the
generalised chain rule shouldn't the summation go from j=1 to n, since
by construction of the functions, the Jacobian matrix of F is a n*m
matrix, the Jacobian for G is a p*n matrix and the Jacobian for the
composition of G and F is a p*m matrix. Hence k:1..m, i:1..p and j:1..n
?
A: Yes, there's a typo. The sum in the displayed equation goes
from j=1 to n. Thanks for spotting this.
Q: Loads of exam questions and solutions available - you're the best.
A: You're welcome.
Q: In the example 3.4. of the notes how do we get that
z=sqrt(r^2-u^2-v^2)?
A: Here, the parametrisation is x=u, y=v and the equation of the
hemisphere is x^2+y^2+z^2 = r^2, so z can be expressed in terms
of x and y and hence u and v as described. This means the surface itself can
be expressed in terms of u and v
Q: Just wanted to thank you for the answers to the questions i've asked:)
also please thank the problem class Phd student, personally I think he
was great!
A: Thanks. I've passed your comment onto Zohar.
Q: Would it be possible for you to do a quick sheet on parametrisation,
the sections in the notes for line and surface integrals are very
short and don't give much help for tackling difficult problems.
A: I don't think producing a shorter version of something you think
is already short would help. Parametrisation is about understanding
how to think about the problem, and it's not purely mechanical.
Q: How would you recommend to revise? Is doing the homeworks (using the
solutions sometimes as some of them are hard) a good method?
A: Always always always revise from the problem sheets. The exam tests
how well you can do problems, not how much of the notes you've
learnt. Learning is different to understanding. Understanding only
comes from doing, not reading. A maths exam will give you 40% from
learning and 60% from understanding.
Q: Could you hijack Thursday's office hour? Could you do it on
parametisation? Thanks for the revision lectures so far.
A: Was away yesterday. Too late to organise timetabling. Sorry.
Q: How do you know which parametrisation to use? I find that the hardest
part of answering questions...
A: That's part of the skill of answering the question. That's all I can say.
Q: Q1b on the revision problem sheet and Q7 on problem sheet 1 are of the
same style, but I do not understand what you are applying the chain
rule to (when calculating the second part) and why it equates to zero.
Could you please clarify this?
A: Too hard really to explain again in text. You have F_1(x,y,u,v) = 0
and then you take d/dx of this but u = u(x,y) and v = v(x,y) so you
need to apply the chain rule. d/dx of zero on the RHS is 0 of course.
Q: Please could you put on one more revision lecture, I've found them
extremely helpful
A: I'm glad you've found them helpful. But all good things must come
to an end. Actually, I don't have any spare time before Friday,
so it's not going to happen unless I hijack Thursday morning's office
hour.
Q: Are you gonna release solutions of revision sessions questions?
A: I am during the sessions. I guess not everyone can make them, so
yes, I'll do this after each session although this requires some
work on my part.
Q: Are questions similar to Q9,10 worksheet 2 likely to appear on the
exam?
A: There's no reason why.
Q: Could you kindly supply answers from problems class week 4? I have the
class notes but cannot seem to understand something so would like to
double check against the answers. Thx
A: Arh ! I don't have any notes for the problems classes as I do
them unscripted. Prob easiest to come and talk to me.
Q: How come the definition of the Divergence is to do dV1/dx1 +dV2/dx2
etc but sometimes you have to divide through by the product of the
normalising constants?
A: The divergence is as you have it. Without examples it's not
possible to answer your question. The only thing I guess is that
the definition of div in curvilinear coordinates involves a
prefactor of 1/(h_1 h_2 h_3), but its appearance is established
in the notes.
Q: Sounds like a stupid question: I notice that according to your
handwriting in lectures you didn't put a underline under the
'inverted triangle' when you write 'curl' or 'divergence', but the phd
student in problem classes underlined it. Is both writing pattern
acceptable in exam?
A: Either is acceptable. The underline is sometimes used to signify
the vector-like quality of the div grad and curl operators.
Q: Will there be any regurgitation of proofs required for the exam? A: Unless I've either indicated either on the notes, or in lectures, that something is non-examinable then it is examinable. But look at the previous exam papers: it's really a "methods" course. I.e. can you solve problems.
Q: Where can i find the solutions for 2015 exam?
A: I'll post them soon -- I wanted students to have a chance to
do the 2015 exam without the temptation of looking at solutions.
Q: Are you having office hours on both Thurs and Friday? Where is your
office?
A: Yes and SM2.7 (if you look carefully at the top of this web page you'll
see the answer to the first Q and if you look at the information page in
Week 1 notes you'll find the answer to the 2nd Q)
Q: In the lecture notes on outline proof of the divergence theorem, you
have stated the volume is 0 < x < a, but then in the integral you are
integrating between a and a rather than a and 0, is this correct?
I mean in the double intergral between 0 < y < b and 0 < z < c, where it says
v1(a,y,z)-v1(a,y,z), should it be v1(a,y,z)-v1(0,y,z) or not?
A: I see. Yes, you are right, there is a typo in the notes and
it should be as you have noted. Thanks for this. I have updated the
week 6 notes.
Q: For the exam, are we expected to know the formulas derived in the
lectures for the transformation of div/curl in curvilinear
coordinates?
A: I would give these formulae. The general rule might be, would
I as a lecturer be able to remember a formula ? If so, I expect you
to know it. That doesn't sound helpful, but I can remember the
formula for the transformation of the gradient. I can also remember
the transformation into spherical polars as well as the divergence
theorem, Stokes theorem, even Green's theorem in the plane. But
some formulae are pretty complicated and used relatively infrequently.
I hope this helps.
Q: Can you provide Marker's feedback on all homework sheets as I found
the feedback for the first homework quite useful?
A: Yes, thanks for spotting that this is not up to date. I'll upload
those I have asap and add the others as they come.
Q: on homework sheet 4, question 2, should the integral not evaluate to
(2/3)*(l^4) not (2/3)*(l^3)?
A: I think you mean Q4 on sheet 4 in which the power of l should be
4 not 3 as stated in the solutions. Thanks for pointing this typo out.
Q: Could you put up the notes as a single document please?
A: Here
Q: Are you able to provide a LaTeX version of the notes?
A: Interesting question... No.
Q: Would you be able to provide a set of printed notes please?
A: I've provided notes online that students can download and print if
they want a printed set of the electronic version of the notes.
Q: For the week 1 printed notes section 1.2, you defined the derivative
as a matrix with size m x n. Surely the size is n x m instead?
A: Yes, you are quite right. Thanks for pointing this out.
Q: Lecture notes aren't sufficient to do the problem sheets especially
question 6 part b of sheet 4.
A: The lecture notes cover all of the material needed to do all of
the questions on all of the sheets. However, students that need
additional help understanding the notes or that might benefit from
further examples could try looking in books on the subject (e.g.
from the reading list on the unit description page)
Q6(b) on sheet 4 involves: taking the curl of F, which is covered in the
notes; doing surface integrals, which is covered in the notes; verifying
Stokes' theorem, which is covered in the notes.
Q: I find the lectures very fast paced and find I spend the majority of
the time trying to write down quickly what you've been writing. So, I
don't get a chance to think about what I'm actually writing and listen
properly to what you are saying. I would find it really useful if the
lecture notes were posted at the beginning of the week so we could go
through them before the lecture and then understand more when you're
lecturing. I know other people on the course feel this way also and
was wondering if you would please do that?
A: No. I understand your problem. I had it too when I was a student.
I found a solution and borrowed someone elses notes after the lecture.
Q: Can you do more examples that are similar to the homework questions,
because everything in lectures makes sense and then when we come to do
the homework we have no idea where to start.
A: In the lectures, the new concepts are always supported by examples.
The problem sheet questions should all be doable if you can follow
the ideas and examples from the class. Some will be harder than others.
Within the course, it would be hard for me to fit in more examples.
The problems classes provide that opportunity and, in going through
exam questions, I am also covering questions which could and sometimes
do appear in exams. Finally, I think the solution sheets provide almost
lecture-quality steps in the answers and should be able to help you
even if you have difficulty in answering the question unsupported.
When I took over this course there were only 3 questions per worksheet
and I have extended the number of questions and the range of difficulty
so that there is more material to practice on.
Q: Could you go over the answers to this weeks homework questions for
Problems 2? Using suffix notation is really hard to understand
A: Lucky for you I hadn't yet decided on material for this afternoon's
Problems Class... I will do some HW Q's in that session
Q: Would it be possible to add a link to this page on Blackboard for
convenience?
A: I had asked for this to be done and was told it had been fixed. I will check again
Q: Where do we pick up our homework? A: Look at the info at the top of this page
Q: Are we required compulsorily to hand exercises on Tuesday to the box? Or it's just a way to help us? A: The homework does not count towards the unit assessment. The unit is assessed by exam only
I'm really enjoying the course. So far I think Richard Porter has been faultless. It's a shame it's only a 6 week course Response: Not for me :)
Q: Is it possible to get the Maths Cafe person's email address ? A: It's Dan Taylor on dt13029.2013 at my.bristol.ac.uk
Q: In the lecture notes of week 1, page 3, section 1.2 the derivative of
the map, the matrix of F'(x) is mxn or nxm? thanks.
A: It should be n by m: there are n components of F and m variables in play.
Q: Is it possible to make the notes available online before the lecture?
A: It's a policy I have not to hand out notes before the lecture. I know this isn't the best solution for everyone, but if I make the notes available it's my experience that it hurts more people than it helps. I will make them available to download at the end of each week.
Q: You're going too fast A: Yes, I'm aware I've set off with some good pace and I will slow it down a bit
Q: Where do I find the homework ? A: We're only 2 lectures in: the homework will be set on Tuesday. I will hand out problem sheets in class and online.
Q: Are you able to hand notes out in lectures? If a lecture is missed, it's difficult to keep up in the subsequent lectures if notes for the previous lecture aren't available until the end of the week A: I'm uploading notes at the end of the week. If you miss a lecture, you could find someone else in class who's got the notes and copy those